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MATH.-STAT. SIM ISAAC MIBWf OM« NEWTON'S PRINCIPIA. THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY, BY SIR ISAAC NEWTON; TRANSLATED INTO ENGLISH BY ANDREW MOTTE. TO WHICH IS ADDKTV NEWTON'S SYSTEM OF THE WORLD ; With a Portrait taken from the Bust in the Royal Observatory at Greenwich. FIRST AMERICAN EDITION, CAREFULLY REVISED AND CORRECTED, WITH A LIFE OF THE AUTHOR, BY PI. W. CHITTENDEN, M. A., &e. NEW-YORK PUBLISHED BY DANIEL ADEE, 45 LIBERTY STREET. p*- Kntered according to Act of Congress, in the year 1846, by DANIEL ADEE. 3!Ltht Clerk's Office ut'tiie Southern Oisli:ct Court of New-York. TWuey * Lockwoof, Stom 16 Spruce St. N. Y. DEDICATION. TO THE TEACHERS OF THE NORMAL SCHOOL OF THE STATE OF NEW-YORK. GENTLEMEN ! A stirring freshness in the air, and ruddy streaks upon the horizon of the moral world betoken the grateful dawning of a new ora. The days of a drivelling instruction are departing. With us is the opening promise of a better time, wherein genuine man hood doing its noblest work shall have adequate reward. TEACHER is the highest and most responsible office man can fill. Its dignity is, and will yet be held commensurate with its duty — a duty boundless as man's intellectual capacity, and great as his moral need — a duty from the performance of which shall emanate an influence not limited to the now and the here, but which surely will, as time flows into eternity and space into infinity, roll up, a measureless curse or a measureless blessing, in inconceivable swellings along the infinite curve. It is an office that should be esteemed of even sacred import in this country. Ere long a hun dred millions, extending from the Atlantic to the Pacific, from Baffin's Bay to that of Panama, shall call themselves American citizens. What a field for those two master-passions of the hu man soul — the love of Rule, and the love of Gain ! How shall our liberties continue to be preserved from the graspings of Am bition and the corruptions of Gold ? Not by Bills of Rights 4 DEDICATION. Constitutions, and Statute Books ; but alone by the rightly culti vated hearts and heads of the PEOPLE. They must themselves guard the Ark. It is yours to tit them for the consecrated charge. Look well to it : for you appear clothed in the majesty of great power ! It is yours to fashion, and to inform , to save, and to perpetuate. You are the Educators of the People : you are the prime Conservators of the public weal. Betray your trust, and the sacred fires would go out, and the altars crumble into dust : knowledge become lost in tradition, and Christian no bleness a fable ! As you, therefore, are multiplied in number, elevated in consideration, increased in means, and fulfill, well and faithfully, all the requirements of true Teachers, so shall our fa voured land lift up her head among the nations of the earth, and call herself blessed. In conclusion, Gentlemen, to you, as the conspicuous leaders in the vast and honourable labour of Educational Helbrm, ana Popular Teaching, the First American Edition of the PRINCIPIA ol Newton — the greatest wrork of the greatest Teacher — is most respectfully dedicated. N. W. CHITTENDEN. INTRODUCTION TO THE AMERICAN EDITION. THAT the PRINCIPIA of Newton should have remained so gen erally unknown in this country to the present day is a somewhat remarkable fact ; because the name of the author, learned with the very elements of science, is revered at every hearth-stone where knowledge and virtue are of chief esteem, while, abroad, in all the high places of the land, the character which that name recalls is held up as the noblest illustration of what MAN may be, and may do, in the possession and manifestation of pre-eminent intellectual and moral worth ; because the work is celebrated, not only in the history of one career and one mind, but in the history of all achievement and human reason itself; because of the spirit of inquiry, which has been aroused, and which, in pursuing its searchings, is not always satisfied with stopping short of the foun tain-head of any given truth ; and, finally, because of the earnest endeavour that has been and is constantly going on, in many sections of the Republic, to elevate the popular standard of edu cation and give to scientific and other efforts a higher and a better aim. True, the PRINCIPIA has been hitherto inaccessible to popular use. A few copies in Latin, and occasionally one in English may be found in some of our larger libraries, or in the possession of some ardent disciple of the great Master. But a d^ad language in the one case, and an enormous price in both, particularly in that of the English edition, have thus far opposed very sufficient obstacles to the wide circulation of the work. It is now, how ever, placed within the reach of all. And in performing this la bour, the utmost care has been taken, by collation, revision, and otherwise, to render the First American Edition the most accurate and beautiful in our language. u Le plus beau monument que l? on puisse clever a la gloire de Newton, c'est une bonne edition de ses ouvrages :" and a monument like unto that we would here V: INTRODUCTION TO set up. The PRINCIPIA, above all, glows with the immortality of a transcendant mind. Marble and brass dissolve and pass away ; but the true creations of genius endure, in time and beyond time, forever : high upon the adamant of the indestructible, they send forth afar and near, over the troublous waters of life, a pure, un wavering, quenchless light whereby the myriad myriads of barques, richly laden with reason, intelligence and various faculty, are guided through the night and the storm, by the beetling shore and the hidden rock, the breaker and the shoal, safely into havens calm and secure. To the teacher and the taught, the scholar and the student, the devotee of Science and the worshipper of Truth, the PRINCIPIA must ever continue to be of inestimable value. If to educate means, not so much to store the memory with symbols and facts, as to bring forth the faculties of the soul and develope them to the full by healthy nurture and a hardy discipline, then, what so effec tive to the accomplishment of that end as the study of Geometri cal Synthesis ? The Calculus, in some shape or other, is, indeed, necessary to the successful prosecution of researches in the higher branches of philosophy. But has not the Analytical encroached upon the Synthetical, and Algorithmic Formulae been employed when not requisite, either for the evolution of truth, or even its apter illustration ? To each method belongs, undoubtedly, an appropriate use. Newton, himself the inventor of Fluxions, censured the handling of Geometrical subjects by Algebraical calculations ; and the maturest opinions which he expressed were additionally in favour of the Geometrical Method. His prefer ence, so strongly marked, is not to be reckoned a mere matter oi taste ; and his authority should bear with preponderating weight upon the decision of every instructor in adopting what may be deemed the best plan to insure the completes! mental develop ment. Geometry, the vigorous product of remote time ; blended with the earliest aspirations of Science and the earliest applica tions of Art ; as well in the measures of music as in the move ment of spheres ; as wholly in the structure of the atom as in that of the world; directing MOTION and shaping APPEARANCE; in a wonl, *t the moulding of the created all, is, in comprehensive THE AMERICAN EDITION. Vll view, the outward form of that Inner Harmony of which and in which all things are. Plainly, therefore, this noble study has other and infinitely higher uses than to increase the power of ab straction. A more general and thorough cultivation of it should oe strenuously insisted on. Passing from the pages of Euclid or Legendre, might not the student be led, at the suitable time, to those of the PRINCIPIA wherein Geometry may be found in varied use from the familiar to the sublime ? The profoundest and the happiest results, it is believed, would attend upon this enlargement of our Educational System. Let the PRINCIPIA, then, be gladly welcomed into every Hall where a TRUE TEACHER presides. And they who are guided to the diligent study of this incomparable work, who become strengthened by its reason, assured by its evidence, and enlight ened by its truths, and who rise into loving communion with the great and pure spirit of its author, will go forth from the scenes of their pupilage, and take their places in the world as strong- minded, right-hearted men — such men as the Theory of our Government contemplates and its practical operation absolutely demands. LIFE OF SIE ISAAC NEWTON. Nec fas est proprius mortal? attingere Divos. — HALLEY. FROM the thick darkness of the middle ages man's struggling spirit emerged as in new birth ; breaking out of the iron control of that period ; growing strong and confident in the tug and din of succeeding conflict and revolution, it bounded forwards and upwards with resistless vigour to the investigation of physical and moral truth ; ascending height after height ; sweeping afar over the earth, penetrating afar up into the heavens ; increasing in en deavour, enlarging in endowment ; every where boldly, earnestly out-stretching, till, in the AUTHOR of the PRINCIPIA, one arose, who, grasping the master-key of the universe and treading its celestial paths, opened up to the human intellect the stupendous realities of the material world, and, in the unrolling of its harmo nies, gave to the human heart a new song to the goodness, wis dom, and majesty of the all-creating, all-sustaining, all-perfect God. Sir Isaac Newton, in whom the rising intellect seemed to attain, as it were, to its culminating point, was born on the 25th of De cember, O. S. 1642 — Christmas day — at Woolsthorpe, in the parish of Colsterworth, in Lincolnshire. His father, John New ton, died at the age of thirty-six, and only a few months after his marriage to Harriet Ayscough, daughter of James Ayscough, oi Rutlandshire. Mrs. Newton, probably wrought upon by the early loss of her husband, gave premature birth to her only and posthumous child, of which, too, from its extreme diminutiveness, she appeared likely to be soon bereft. Happily, it was otherwise decreed ! The tiny infant, on whose little lips the breath of life 10 LIFE OF SIR ISAAC NEWTON. so doubtingly hovered, lived ; — lived to a vigorous maturity, to a hale old age ; — lived to become the boast of his country, the won der of his time, and the "ornament of his srjecies." Beyond the grandfather, Robert Newton, the descent of Sir Isaac cannot with certainty be traced. Two traditions were held in the family : one, that they were of Scotch extraction ; the other, that they came originally from Newton, in Lancashire, dwelling, for a time, however, at Westby, county of Lincoln, be fore the removal to and purchase of Woolsthorpe — about a hundred years before this memorable birth. The widow Newton was left with the simple means of a com fortable subsistence. The Woolsthorpe estate together with small one which she possessed at Sewstern, in Leicestershire, yield ed her an income of some eighty pounds ; and upon this limited sum, she had to rely chiefly for the support of herself, and the educa tion of her child. She continued his nurture for three years, when, marrying again, she confided the tender charge to the care of her own mother. Great genius is seldom marked by precocious development ; and young Isaac, sent, at the usual age, to two day schools at Skillington and Stoke, exhibited no unusual traits of character. In his twelfth year, he was placed at the public school at Gran- tham, and boarded at the house of Mr. Clark, an apothecary. But even in this excellent seminary, his mental acquisitions con tinued for a while unpromising enough : study apparentlv had no charms for him ; he was very inattentive, and ranked low in the school. One day, however, the boy immediately above our seem ingly dull student gave him a severe kick in the stomach ; Isaac, deeply affected, but with no outburst of passion, betook himself, with quiet, incessant toil, to his books ; he quickly passed above the offending classmate ; yet there he stopped not ; the strong spirit was, for once and forever, awakened, and, yielding to itb noble impulse, he speedily took up his position at the head of all. His peculiar character began now rapidly to unfold itself. Close application grew to be habitual. Observation alternated with reflection. " A sober, silent, thinking lad," yet, the wisest and the kindliest, the indisputable leader of his fellows. Gener- LIFE OF SIR ISA VC NEWTON. 11 osity, modesty, and a love of truth distinguished him then as ever afterwards. He did not often join his classmates in play ; but he would contrive for them various amusements of a scientific kind. Paper kites he introduced ; carefully determining their best form and proportions, and the position and number of points whereby to attach the string. He also invented paper lanterns ; these served ordinarily to guide the way to school in winter mornings, but occasionally for quite another purpose ; they were attached to the tails of kites in a dark night, to the dismay of the country people dreading portentous comets, and to the immeasureable delight ol his companions. To him, however, young as he was, life seemed to have become an earnest thing. When not occupied with his studies, his mind would be engrossed with mechanical contrivances ; now imitating, now inventing. He became singularly skilful in the use of his little saws, hatchets, hammers, and other tools. A windmill was erected near Grantham ; during the operations ol the workmen, he was frequently present ; in a short time, he had completed a perfect working model of it, which elicited general admiration. Not content, however, with this exact imitation, he conceived the idea of employing, in the place of sails, animal power , and, adapting the construction of his mill accordingly, he enclosed in it a mouse, called the miller, and which by acting on a sort ot treadvvheel, gave motion to the machine. He invented, too, a mechanical carriage — having four wheels, and put in motion with a handle worked by the person sitting inside. The measurement of time early drew his attention. He h'rst constructed a water clock, in proportions somewhat like an old-fashioned house clock. The index of the dial plate was turned by a piece of wood acted upon by dropping water. This instrument, though long used by himself, and by Mr. Clark's family, did not satisfy his inquiring mind. His thoughts rose to the sun ; and, by careful and oft-re peated observations of the solar movements, he subsequently formed many dials. One of these, named Isaac's dial, was the accurate result of years' labour, and was frequently referred to for the hour of the day by the country people. May we not discern in these continual efforts — the diligent re search^ the patient meditation, the aspiring glance, and the energy 12 LIFE OF SIR ISAAC NEWTON. of discovery — the stirring elements of that wondrous spirit, which, clear, calm, and great, moved, in after years, through deep onward through deep of Nature's mysteries, unlocking her strongholds, dispelling darkness, educing order — everywhere si lently conquering. Newton had an early and decided taste for drawing. Pictures, taken sometimes from copies, but often from life, and drawn, coloured and framed by himself, ornamented his apartment. He was skilled also, in poetical composition, " excelled in making verses ;" some of these were borne in remembrance and repeated, seventy years afterward, by Mrs. Vincent, for whom, in early youth, as Miss Storey, he formed an ardent attachment. She was the sister of a physician resident near Woolsthorpe ; but Newton's intimate acquaintance with her began at Grantham. where they were both numbered among the inmates of the same house. Two or three years younger than himself, of great per sonal beauty, and unusual talent, her society afforded him the greatest pleasure ; and their youthful friendship, it is believed, gradually rose to a higher passion ; but inadequacy of fortune prevented their union. Miss Storey was afterwards twice mar ried ; Newton, never; his esteem for her continued unabated during life, accompanied by numerous acts of attention and kindness. In 1656, Newton's mother was again left a widowr, and took up her abode once more at Woolsthorpe. He was now fifteen years of age, and had made great progress in his studies ; but she, desirous of his help, and from motives of economy, recalled him from school. Business occupations, however, and the manage ment of the farm, proved utterly distasteful to him. When sent to Grantham Market on Saturdays, he would betake himself to his former lodgings in the apothecary's garret, where some of Mr. Clark's old books employed his thoughts till the aged and trust worthy servant had executed the family commissions and announced the necessity of return : or, at other times, our young philosopher would seat himself under a hedge, by the wayside, and continue his studies till the same faithful personage — proceeding alone to the town and completing the day's business — stopped as he re- LIFE OF SIR ISAAC NEWTON, 13 turned. The more immediate affairs of the farm received no better attention. In fact, his passion for study grew daily more absorbing, and his dislike for every other occupation more in tense. His mother, therefore, wisely resolved to give him all the advantages which an education could confer. He was sent back to Grantham school, where he remained for some months in busy preparation for his academical studies. At the recommendation of one of his uncles, who had himself studied at Trinity College, Cambridge, Newton proceeded thither, and was duly admitted. on the 5th day of June 1660, in the eighteenth year of his age. The eager student had now entered upon a new and wider field ; and we find him devoting himself to the pursuit of know ledge with amazing ardour and perseverance. Among other sub jects, his attention was soon drawn to that of Judicial Astrology He exposed the folly of this pseudo-science by erecting a figure with the aid of one or two of the problems of Euclid ; — and thus began his study of the Mathematics. His researches into this science were prosecuted with unparallelled vigour and success. Regarding the propositions contained in Euclid as self-evident truths, he passed rapidly over this ancient system — a step which he afterward much regretted — and mastered, without further pre paratory study, the Analytical Geometry of Descartes. Wallis's Arithmetic of Infinites, Saunderson's Logic, and the Optics of Kepler, he also studied with great care ; writing upon them many comments ; and, in these notes on Wallis's work was un doubtedly the germ of his fluxionary calculus. His progress was so great that he found himself more profoundly versed than his tutor in many branches of learning. Yet his acquisitions were not gotten with the rapidity of intuition ; but they were thoroughly made and firmly secured. Quickness of apprehension, or Intel lectual nimbleness did not belong to him. He saw too far : his, insight was too deep. He dwelt fully, cautiously upon the least subject ; while to the consideration of the greatest, he brought a massive strength joined with a matchless clearness, that, regard less of the merely trivial or unimportant, bore with unerring sa gacity upon the prominences of the subject, and, grappling with its difficulties, rarely failed to surmount them. 14 LIFE OF SIR ISAAC NEWTON His early and fast friend, Dr. Barrow — in compass of inven tion only inferior to Newton — who had been elected Professor of Greek in the University, in 1660, was made Lucasian Profes sor of Mathematics in 1663, and soon afterward delivered his Optical Lectures : the manuscripts of these were revised by New ton, and several oversights corrected, and many important sug gestions made by him ; but they were not published till 1669. In the year 1665, he received the degree of Bachelor of Arts ; and, in 1666, he entered upon those brilliant and imposing dis coveries which have conferred inappreciable benefits upon science, and immortality upon his own name. Newton, himself, states that he was in possession of his Method of Fluxions, " in the year 1666, or before." Infinite quantities had long been a subject of profound investigation ; among the ancients by Archimedes, and Pappus of Alexandria ; among the moderns by Kepler, Cavaleri, Roberval, Fermat and Wallis. With consummate ability Dr. Wallis had improved upon the la- hours of his predecessors : with a higher power, Newton moved forwards from where Wallis stopped. Our author first invented his celebrated BINOMIAL THEOREM. And then, applying this Theorem to the rectification of curves, and to the determination of the surfaces and contents of solids, and the position of their centres of gravity, he discovered the general principle of deducing the areas of curves from the ordinate, by considering the area as a nascent quantity, increasing by continual fluxion in the propor tion of the length of the ordinate, and supposing the abscissa to increase uniformly in proportion to the time. Regarding lines as generated by the motion of points, surfaces by the motion of lines, and solids by the motion of surfaces, and considering that the ordinates, abscissae, &c., of curves thus formed, vary accord ing to a regular law depending on the equation of the curve, he deduced from this equation the velocities with which these quantities are generated, and obtained by the rules of infinite series, the ultimate value required. To the velocities with which every line or quantity is generated, he gave the name of FLUX IONS, and to the lines or quantities themselves, that of FLUENTS. A discovery that successively baffled the acutest and strongest LIFE OF SIR ISAAC NEWTON. 15 intellects : — that, variously modified, has proved of incalculable service in aiding to develope the most abstruse and the highest 'ruths in Mathematics and Astronomy : and that was of itself enough to render any name illustrious in the crowded Annals of Science. At this period, the most distinguished philosophers were direct ing all their energies to the subject of light and the improvement of the refracting telescope. Newton, having applied himself to the grinding of "optic glasses of other figures than spherical," ex perienced the impracticability of executing such lenses ; and con jectured that their defects, and consequently those of refracting telescopes, might arise from some other cause than the imperfect convergency of rays to a single point. He accordingly "procured a triangular glass prism to try therewith the celebrated phenom ena of colours." His experiments, entered upon with zeal, and conducted with that industry, accuracy, and patient thought, for which he was so remarkable, resulted in the grand conclusion, that LIGHT WAS NOT HOMOGENEOUS, BUT CONSISTED OF RAYS, SOME OF WHICH WERE MORE REFRANGIBLE THAN OTHERS. This profound and beautiful discovery opened up a new era in the History of Optics. As bearing, however, directly upon the construc tion of telescopes, he saw that a lens refracting exactly like a prism would necessarily bring the different rays to different foci, at different distances from the glass, confusing and rendering the vision indistinct. Taking for granted that all bodies produced spectra of ^ jtial length, he dismissed all further consideration of the refracting instrument, and took up the principle of reflection. Rays of all colours, he found, were reflected regularly, so that the angle of reflection was equal to the angle of incidence, and hence he concluded that ojitical instruments might be brought to any degree of perfection imaginable, provided reflecting specula of the requisite figure and finish could be obtained. At this stage of his optical researches, he was forced to leave Cambridge on account of the plague which was then desolating England. He retired to Woolsthorpe. The old manor-house, in which he was born, was situated in a beautiful little valley, on the west side of the river Witham ; and here in the quiet home of his boyhood, 2 16 LIFE OF SIR ISAAC NEWTON. he passed his days in serene contemplation, while the stalking pestilence was hurrying its tens of thousands into undistinguisha • ble graves. Towards the close of a pleasant day in the early autumn of 1666, he was seated alone beneath a tree, in his garden, absorbed in meditation. He was a slight young man ; in the twenty-fourth year of his age ; his countenance mild and full of thought. For a century previous, the science of Astronomy had advanced with rapid strides. The human mind had risen from the gloom and bondage of the middle ages, in unparalleled vigour, to unfold the system, to investigate the phenomena, and to establish the laws of the heavenly bodies. Copernicus, Tycho Brahe, Kepler, Galileo, and others had prepared and lighted the way for him who was ta give to their labour its just value, and to their genius its true lustre. At his bidding isolated facts were to take order as parts of one harmonious whole, and sagacious conjectures grow luminous in the certain splendour of demonstrated truth. And this ablest man had come — was here. His mind, familiar with the knowledge of past effort, and its unequalled faculties develop ed in transcendant strength, was now moving on to the very threshold of Its grandest achievement. Step by step the untrod den path was measured, till, at length, the entrance seemed dis closed, and the tireless explorer to stand amid the first opening wonders of the universe. The nature of gravity — that mysterious power which causes all bodies to descend towards the centre of the earth — had, in deed, dawned upon him. And reason busily united link to link of that chain which was yet to be traced joining the least to the vastest, the most remote to the nearest, in one harmonious bond. From the bottoms of the deepest caverns to the summits of the highest mountains, this power suffers no sensible change : may not its action, then, extend to the moon ? Undoubtedly : and furthei reflection convinced him that such a power might be .sufficient for retaining that luminary in her orbit round the earth. But, though this power suffers no sensible variation, in the little change of distance from the earth's centre, at which we may place our- .'«lves, yet, at the distance of the moon, :miy not its force undergo LIFE OF SIR ISAAC NEWTON. 17 more or less diminution ? The conjecture appeared most proba ble : and, in order to estimate what the degree of diminution might be, he considered that if the moon be retained in her orbit by the force of gravity, the primary planets must also be carried round the sun by the like power; and, by comparing the periods of the several planets with their distances from the sun, he found that, if they were held in their courses by any power like gravity, its strength must decrease in the duplicate proportion of the in crease of distance. In forming this conclusion, he supposed the planets to move in perfect circles, concentric to the sun. Now was this the law of the moon's motion ? Was such a force, em anating from the earth and directed to the moon, sufficient, when diminished as the square of the distance, to retain her in her orbit ? To ascertain this master-fact, he compared the space through which heavy bodies fall, in a second of time, at a given distance from the centre of the earth, namely, at its surface, with the space through which the moon falls, as it were, to the earth, in the same time, while revolving in a circular orbit. He was absent from books ; and, therefore, adopted, in computing the earth's diameter, the common estimate of sixty miles to a degree of latitude as then in use among geographers and navigators. The result of his calculations did not, ot course, answer his ex pectations ; hence, he concluded that some other cause, beyond the reach of observation — analogous, perhaps, to the vortices of Des cartes — joined its action to that of the power of gravity upon the rnooil. Though by no means satisfied, he yet abandoned awhile further inquiry, and remained totally silent upon the subject. These rapid marches in the career of discovery, combined with the youth of Newton, seem to evince a penetration the most lively, and an invention the most exuberant. But in him there was a conjunction of influences as extraordinary as fortunate. Study, unbroken, persevering and profound carried on its inform ing and disciplining work upon a genius, natively the greatest, and rendered freest in its movements, and clearest in its vision, through the untrammelling and enlig} tenirig power of religion. And, in this happy concurrence, are to be sought the elements of those amazing abilities, which, grasping, with equal facility, the 18 LIFE OF SIR ISAAC NEWTON. minute and the stupendous, brought these successively to light, and caused science to make them her own. In 1667, Newton was made a Junior Fellow ; and, in the year following, he took his degree of Master of Arts, and was appoint ed to a Senior Fellowship. On his return to Cambridge, in 1668, he resumed his optical labours. Having thought of a delicate method of polishing metal, he proceeded to the construction of his newly projected reflect ing telescope ; a small specimen of which he actually made with his own hands, It was six inches long ; and magnified about forty times ; — a power greater than a refracting instrument of six feet tube could exert with distinctness. Jupiter, with his four satellites, and the horns, or moon-like phases of Venus were plainly visible through it. THIS WAS THE FIRST REFLECTING TELESCOPE EVER EXECUTED AND DIRECTED TO THE HEAVENS. He gave an account of it, in a letter to a friend, dated February 23d, 1668-9 — a letter which is also remarkable for containing the firs' allusion to his discoveries " concerning the nature of light." En couraged by the success of his first experiment, he again executed with his own hands, not long afterward, a second and superior instrument of the same kind. The existence of this having come to the knowledge of the Royal Society of London, in 1671, they requested it of Newton for examination. He accordingly sent it to them, It excited great admiration; it was shown to the king* a drawing and description of it was sent to Paris ; and the tele- scope itself was carefully preserved in the Library of the Society. Newton lived to see his invention in public use, and of eminent service in the cause of science. In the spring of 1669, he wrote to his friend Francis Aston, Esq., then about setting out on his travels, a letter of advice and directions, it was dated May 18th, and is interesting as exhibit ing some of the prominent features in Newton's character. Thus : — " Since in your letter you give me so much liberty of spending my judgment about what may be to your advantage in travelling, 1 shall do it more freely than perhaps otherwise would have been decent, Fir,ct, then, I will lay down some general rules, most of LIFE OF SIR ISAAC NEWTON. 19 which, I bolieA e, you have considered already ; but if any of them be new to you, they may excuse the rest ; if none at all, yet is my punishment more in writing than yours in reading. "When you come into any fresh company. 1. Observe their humours. 2. Suit your own carriage thereto, by which insinua tion you will make their converse more free and open. 3. Let your discourse be more in queries and doubtings than peremptory assertions or disputings, it being the design of travellers to learn, not to teach. Besides, it will persuade your acquaintance that you have the greater esteem of them, and so make them more ready to communicate what they know to you ; whereas nothing sooner occasions disrespect and quarrels than peremptoriness. You will find little or no advantage in seeming wiser or much more ignorant than your company. 4. Seldom discommend any thing though never so bad, or do it but moderately, lest you be unexpectedly forced to an unhandsome retraction. It is safer to commend any thing more than it deserves, than to discommend a thing so much as it deserves; for commendations meet not so often with oppositions, or, at least, are not usually so ill re sented by men that think otherwise, as discommendations ; and you will insinuate into men's favour by nothing sooner than seem ing to approve and commend what they like ; but beware o doing it by comparison. 5. If you be affronted, it is better, in c foreign country, to pass it by in silence, and with a jest, though with some dishonour, than to endeavour revenge ; for, in the first case, your credit's ne'er the worse when you return into England, or come into other company that have not heard of the quarrel. But, in the second case, you may bear the marks of the quarrel while you live, if you outlive it at all. But, if you find yoursell unavoidably engaged, 'tis best, I think, if you can command your passion and language, to keep them pretty evenly at some certain moderate pitch, not much heightening them to exasperate your adversary, or provoke his friends, nor letting them grow overmuch dejected to make him insult. In a word, if you can keep reason above passion, that and watchfulness will be your best defendants. To which purpose you may consider, that, though such excuses is this — He provok't me so much I could not forbear — may pass 20 LIFE OF SIR ISAAC NEWTON. among friends, yet amongst strangers they are insignificant, ina only argue a traveller's weakness. " To these I may add some general heads for inquiries or ob servations, such as at present I can think on. As, 1. To observe the policies, wealth, and state affairs of nations, so far as a soli- fary traveller may conveniently do. 2. Their impositions upon all sorts of people, trades, or commodities, that are remarkable. 3. Their laws and customs, how far they differ from ours. 4. Their trades and arts wherein they excel or come short of us in England. 5. Such fortifications as you shall meet with, their fashion, strength, and advantages for defence, and other such mili tary affairs as are considerable. 6. The power and respect be longing to their degrees of nobility or magistracy. 7. It will not be time misspent to make a catalogue of the names and excellen cies of those men that are most wise, learned, or esteemed in any nation. 8. Observe the mechanism and manner of guiding ships. 9. Observe the products of Nature in several places, especially in mines, with the circumstances of mining and of extracting metals or minerals out of their ore, and of refining them ; and if you meet with any transmutations out of their own species into another (as out of iron into copper, out of any metal into quick silver, out of one salt into another, or into an insipid body, &c.), those, above all, will be worth your noting, being the most lucif- erous, and many times lucriferous experiments, too, in philosophy. 10. The prices of diet and other things. 11. And the staple commodities of places. " These generals (such as at present I could think of), if they will serve for nothing else, yet they may assist you in drawing up a model to regulate your travels by. ' As for particulars, these that follow are all that 1 can now think of, viz. ; whether at Schem- nitium, in Hungary (where there are mines of gold, copper, iron, vitriol, antimony, &c.). they change iron into copper by dissolving »t in a vitriolate water, which they find in cavities of rocks in the mines, and then melting the slimy solution in a stroi ig fire, which in the cooling proves copper. The like is said to be done in other places, which I cannot now remember ; perhaps, too, it may be lone in Italy. For about twenty or thirty years agone there was LIFE OF SIR ISAAC NEWTON, 21 a certain vitriol came from thence (called Roman vitriol), but of a nobler virtue than that which is now called by that name ; which vitriol is not now to be gotten, because, perhaps, they make a greater gain by some such trick as turning iron into copper with it than by selling it. 2. Whether, in Hungary, Sclavonia, Bohemia, near the town Eila, or at the mountains of Bohemia near Silesia, there be rivers whose waters are impregnated with gold ; perhaps, the gold being dissolved by some corrosive water like aqua regis, and the solution carried along with the stream, that runs through the mines. And whether the practice of laying mercury in the rivers, till it be tinged with gold, and then strain ing the mercury through leather, that the gold may stay behind, be a secret yet, or openly practised. 3. There is newly con trived, in Holland, a mill to grind glasses plane withal, and I think polishing them too ; perhaps it will be worth the while to see it. 4. There is in Holland one Borry, who some years since was imprisoned by the Pope, to have extorted from him secrets (as I am told) of great worth, both as to medicine and profit, but he escaped into Holland, where they have granted him a guard. I think he usually goes clothed in green. Pray inquire what you can of him, and whether his ingenuity be any profit to the Dutch. You may inform yourself whether the Dutch have any tricks to keep their ships from being all worm-eaten in their voyages to the Indies. Whether pendulum clocks do any service in finding out the longitude, &c. " I am very weary, and shall not stay to part with a long compliment, only I wish you a good journey, and God be with you." It was not till the month of June, 1669, that our author made known his Method of Fluxions. He then communicated the work which he had composed upon the subject, and entitled, ANALYSIS PER EQUATIONES NUMERO TERMINORUM INFINITAS, to his friend Dr. Barrow. The latter, in a letter dated 20th of the same month, mentioned it to Mr. Collins, and transmitted it to him, on the 31st of July thereafter. Mr. Collins greatly approv> ed of the work ; took a copy of it ; and sent the original back to Dr. Barrow. During the same and the two following years, Mr < LIFE OF SIR ISAAC NEWTON. Collins, by his extensive correspondence, spread the knowledge of this discovery among the mathematicians in England, Scotland, France, Holland and Italy. Dr. Barrow, having resolved to devote himself to Theology, resigned the Lucasian Professorship of Mathematics, in 1669, in favour of Newton, who accordingly received the appointment to the vacant chair. During the years 1669, 1670, and 1671, our author, as such Professor, delivered a course of Optical Lectures. Though these contained his principal discoveries relative to the different re- frangibility of light, yet the discoveries themselves did not be come publicly known, it seems, till he communicated them to the Royal Society, a few weeks after being elected a member there of, in the spring of 1671-2. He now rose rapidly in reputation, and was soon regarded as foremost among the philosophers of the age. His paper on light excited the deepest interest in the Royal Society, who manifested an anxious solicitude to secure the author from the " arrogations of others," and proposed to publish his discourse in the monthly numbers in which the Transactions were given to the world. Newton, gratefully sensible of these expres sions of esteem, willingly accepted of the proposal for publication. He gave them also, at this time, the results of some further ex periments in the decomposition and re-composition of light : — that the same degree of refrangibility always belonged to the same colour, and the same colour to the same degree of refrangibility : that the seven different colours of the spectrum were original, or simple, and that whiteness^ or white light was a compound of all these seven colours. The publication of his new doctrines on light soon called forth violent opposition as to their soundness. Hooke and Huygens — men eminent for ability and learning — were the most conspicuous of the assailants. And though Newton effectually silenced all his adversaries, yet he felt the triumph of little gain in comparison .vith the loss his tranquillity had sustained. He subsequently re- narked in allusion to this controversy — and to one with whom he was destined to have a longer and a bitterer conflict — " I was so persecuted with discussions arising from the publication of m v LIFE OF SIR ISAAC NEWTON. 23 theory ot light, that I blamed my own imprudence for parting with so substantial a blessing as rny quiet to run after a shadow.7' In a communication to Mr. Oldenburg, Secretary of the Royal Society, in 1672, our author stated many valuable suggestions re lative to the construction of REFLECTING MICROSCOPES which he considered even more capable of improvement than telescopes. He also contemplated, about the same time, an edition of Kirick- huysen's Algebra, with notes and additions; partially arranging, as an introduction to the work, a treatise, entitled, A Method of Fluxions ; but he finally abandoned the design. This treatise, however, he resolved, or rather consented, at a late period of his life, to put forth separately ; and the plan would probably have been carried into execution had riot his death intervened. It was translated into English, and published in 1736 by John Colson, Professor of Mathematics in Cambridge. Newton, it is thought, made his discoveries concerning the INFLECTION and DIFFRACTION of light before 1674. The phe nomena of the inflection of light had been first discovered more than ten years before by Grimaldi. And Newton began by re peating one of the experiments of the learned Jesuit — admitting a beam of the sun's light through a small pin hole into a dark chamber : the light diverged from the aperture in the form of a, cone, and the shadows of all bodies placed in this light were larger than might have been expected, and surrounded with three coloured fringes, the nearest being widest, and the most remote the narrowest. Newton, advancing upon this experiment, took exact measures of the diameter of the shadow of a human hair, and of the breadth of the fringes, at different distances behind it, and discovered that these diameters and breadths were not pro portional to the distances at which they were measured. He hence supposed that the rays which passed by the edge of the hair were deflected or turned aside from it, as if by a repulsive force, the nearest rays suffering the greatest, the more remote a less degree of deflection. In explanation of the coloured fringes, he queried : whether the rays which differ in refrangibility do not differ also in flexibility, and whether they are n<»t, by these dif ferent inflections, separated from one another, so as after separa- < LIFE OF SIR ISAAC NEWTON. tion to make the colours in the three fringes above described ? Also, whether the rays, in passing by the edges and sides ol bodies, are not bent several times backwards and forwards with an eel-like motion — the three fringes arising from three such bendings ? His inquiries on this subject were here interrupted and fiever renewed. His Theory of the COLOURS of NATURAL BODIES was commu nicated to the Royal Society, in February, 1675. This is justly regarded as one of the profoundest of his speculations. The fun damental principles of the Theory in brief, are : — That bodies possessing the greatest refractive powers reflect the greatest quantity of light ; and that, at the confines of equally refracting media, there is no reflection. That the minutest particles of al most all natural bodies are in some degree transparent. That between the particles of bodies there are pores, or spaces, either empty or filled with media of a less density than the particles themselves. That these particles, and pores or spaces, have some definite size. Hence he deduced the Transparency, Opacity, and colours of natural bodies. Transparency arises from the particles and their pores being too small to cause reflection at their com mon surfaces — the light all passing through ; Opacity from the opposite cause of the particles and their pores being sufficiently large to reflect the light which is " stopped or stifled7' by the multitude of reflections ; and colours from the particles, accord ing to their several sizes, reflecting rays of one colour and trans mitting those of another — or in other words, the colour that meets the eye is the colour reflected, while all the other rays are transmitted or absorbed. Analogous in origin to the colours of natural bodies, he con sidered the COLOURS OF THIN PLATES. This subject was interest ing and important, and had attracted considerable investigation. He, however, was the first to determine the law of the produc tion of these colours, arid, during the same year made known the results of his researches herein to the Royal Society. His mode of procedure in these experiments was simple and curious. He placed a double convex lens of a large known radius of curvature, the flat surface of a plano-convex object glass. Thus, from UFE OF SIR ISAAC NEWTON. 25 their point of contact at the centre, to the circumference of the lens, he obtained plates of air, or spaces varying from the ex- tremest possible thinness, by slow degrees, to a considerable thick ness. Letting the light fall, every different thickness of this plate of air gave different colours — the point of contact of the lens and glass forming the centre of numerous concentric colored nags. Now the radius of curvature of the lens being known, the thickness of the plate of air, at any given point, or where any par ticular colour appeared, could be exactly determined. Carefully noting, therefore, the order in which the different colours ap peared, he measured, with the nicest accuracy, the different thick* nesses at which the most luminous parts of the rings were pro duced, whether the medium were air, water, or mica — all these substances giving the same colours at different thicknesses ; — the ratio of which he also ascertained. From the phenomena obser ved in these experiments, Newton deduced his Theory of Fits of EASY REFLECTION AND TRANSMISSION of light. It consists in suppos ing that every particle of light, from its first discharge from a lumi nous body, possesses, at equally distant intervals, dispositions to be reflected from, or transmitted through the surfaces of bodies upon which it may fall. For instance, if the rays are in a Fit of Easy Reflection, they are on reaching the surface, repelled, thrown off] or reflected from it ; if, in a Fit of Easy Transmission, they are attracted, drawn in, or transmitted through it. By this Theory of Fits, our author likewise explained the colours of thick plates. He regarded light as consisting of small material particles emitted from shining substances. He thought that these parti cles could be re-combined into solid matter, so that " gross bodies and light, were convertible into one another ;" that the particles of light and the particles of solid bodies acted mutually upon each other ; those of light agitating and heating those of solid bodies, and the latter attracting and repelling the former. Newton was the first to suggest the idea of the POLARIZATION of light. In the paper entitled An Hypothesis Explaining Properties of Light, December, 1675, our author first introduced his opinions re specting Ether — opinions which he afterward abandoned and again 26 LIFE OF SIR S.\AC 1SEWTON. permanently resumed — " A most subtle spirit which pervades" ah bodies, and is expanded through all the heavens. It is electric, and almost, if not quite immeasurably elastic and rare. " By the force and action of which spirit the particles of bodies mutually attract one another, at near distances, and cohere, if contiguous ; and electric bodies operate at greater distances, as well repelling as attracting the neighbouring corpuscles ; and light is emitted, -reflected, refracted, inflected and heats bodies ; and all sensation is excited, and the members of animal bodies move at the com mand of the will, namely, by the vibrations of this spirit, mutu ally propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles." This " spirit" was no anima mundi ; nothing further from the thought of Newton ; but was it not, on his part, a par tial recognition of, or attempt to reach an ultimate material force, or primary element, by means of which, " in the roaring loom of time," this material universe, God's visible garment, may be woven for us ? The Royal Society were greatly interested in the results of some experiments, which our author had, at the same time, com municated to them relative to the excitation of electricity in glass ; and they, after several attempts and further direction from him, succeeded in re-producing the same phenomena. One of the most curious of Newton's minor inquiries related to the connexion between the refractive powers and chemical com position of bodies. He found on comparing the refractive powers and the densities of many different substances, that the former were very nearly proportional to the latter, in the same bodies. Unctuous and sulphureous bodies were noticed as remarkable excep tions — as well as the diamond — their refractive powers being two or three times greater in respect of their densities than in the case of other substances, while, as among themselves, the one was generally proportional to the other. He hence inferred as to the diamond a great degree of combustibility •; — a conjecture which the experiments of modern chemistry have shown to be true. The chemical researches of our author were probably pursued with more or less diligence from the time of his witnessing some LIFE OF .SIR ISAAC NEWTON. 27 ?t' the uractical operations in that science at the Apothecary's at Grantham. DE NATURA ACIDORUM is a short chemical paper, on various topics, and published in Dr. Horsley's Edition of his works. TABULA QUANTITATUM E r GRADUUM COLORIS was in serted iii the Philosophical Transactions ; it contains a compara tive scale of temperature from that of melting ice to that of a small kitchen coal-fire. He regarded fire as a body heated so hot as to emit light copiously ; and flame as a vapour, fume, or ex halation heated so hot as to shine. To elective attraction, by the operation of which the small particles of bodies, as he con ceived, act upon one another, at distances so minute as to escape observation, he ascribed all the various chemical phenomena ot precipitation, combination, solution, and crystallization, and the mechanical phenomena of cohesion and capillary attraction. New ton's chemical views were illustrated and confirmed, in part, at least, in his own life-time. As to the structure of bodies, he was of opinion " that the smallest particles of matter may cohere by the strongest attractions, and compose bigger particles of weaker virtue ; and many of these may cohere and compose bigger par tides whose virtue is still weaker ; and so on for divers succes sions, until the progression end in the biggest particles, on which the operations in chemistry and the colours of natural bodies de pend, and which by adhering, compose bodies of sensible magni tude." There is good reason to suppose that our author was a diligent student of the writings of Jacob Behmen ; and that in conjunction with a relative, Dr. Newton, he was busily engaged, for several months in the earlier part of life, in quest of the philosopher's tincture. " Great Alchymist," however, very imperfectly de scribes the character of Behmen, whose researches into things material and things spiritual, things human and things divine, ai- ford the strongest evidence of a great and original mind. More appropriately here, perhaps, than elsewhere, may be given Newton's account of some curious experiments, made in his own person, on the action of light upon the retina, Locke, who was an intimate friend of our author, wrote to him for his opinion on a certain fact stated in Boyle's Book of Colours. Newton, in 2S LIFE OF SIR ISAAC NEWTON. his reply, dated June 30th, 16(Jl, narrates the following circum stances, which probably took place in the course of his optical researches. Thus : — " The observation you mention in Mr. Boyle's Book of Colours I once tried upon myself with the hazard of my eyes. The manner was this ; I looked a very little while upon the sun in the looking-glass with my right eye, and then turned my eyes into a dark corner of my chamber, arid winked, to observe the impres sion made, and the circles of colours which encompassed it, and how they decayed by degrees, and at last vanished. This I re peated a second and a third time. At the third time, when the phantasm of light and colours about it were almost vanished, in tending my fancy upon them to see their last appearance, I found, to my amazement, that they began to return, and by little and little to become as lively and vivid as when I had newly looked upon the sun. But when I ceased to intend my fancy upon them, they vanished again. After this, I found, that as often as I went into the dark, and intended my mind upon them, as when a man looks earnestly to see anything which is difficult to be seen, I could make the phantasm return without looking any more upon the sun ; and the oftener I made it return, the more easily I could make it return again. And, at length, by repeating this, without looking any more upon the sun, I made such an impression on my eye, that, if I looked upon the clouds, or a book, or any bright object, I saw upon it a round bright spot of light like the sun, and, which is still stranger, though I looked upon the sun with my right eye only, and not with my left, yet my fancy began *o make an impression upon my left eye, as well us upon my right. For if I shut my right eye, or looked upon a book, or the clouds, with my left eye, I could see the spectrum of the sun almost as plain as with my right eye, if I did but intend my fancy a little while upon it ; for at first, if I shut my right eye, and looked with my left, the spectrum of the sun did not appear till I intended my fancy upon it ; but by repeating, this appeared every time more easily. And now, in a few hours' time, I had brought my eyes to such a pass, that I could look upon no blight object with either eye, but I saw the sun before me, so that I durst neither write LIFE OF SIR ISAAC NEWTON. 29 nor read ; but to recover the use of my eyes, shut myself up in my chamber made dark, for three days together, and used all means to divert my imagination from the sun. For if I thought upon him, I presently saw his picture, though I was in the dark. But by keeping in the dark, and employing my mind about other things, I began in three or four days to have some use of my eyes again ; and by forbearing to look upon bright objects, recovered them pretty well, though not so well but that, for some months after, the spectrum of the sun began to return as often as I began to meditate upon the phenomena, even though I lay in bed at mid night with my curtains drawn. But now I have been very well for many years, though I am apt to think, if I durst venture my eyes, I could still make the phantasm return by the power of my fancy. This story I tell you, to let you understand, thaj; in the observation related by Mr. Boyle, the man's fancy probably con curred with the impression made by the sun's light to produce that phantasm of the sun which he constantly saw in bright ob jects. And so your question about the cause of phantasm in volves another about the power of fancy, which I must confess is too hard a knot for me to untie. To place this effect in a constant motion is hard, because the sun ought then to appear perpetually. It seems rather to consist in a disposition of the sensorium to move the imagination strongly, and to be easily moved, both by the imagination and by the light, as often as bright objects are looked upon."J Though Newton had continued silent, yet his thoughts were by no means inactive upon the vast subject of the planetary mo tions. The idea of Universal Gravitation, first caught sight of, so to speak, in the garden at Woolsthorpe, years ago, had gradually expanded upon him. We find him, in a letter to Dr. Hooke, Secretary of the Royal Society, dated in November, 1679, pro posing to verify the motion of the earth by direct experiment, namely, by the observation of the path pursued by a body falling from a considerable height. He had concluded that the path would be spiral ; but Dr. Hooke maintained that it would be an eccentric ellipse iu vacuo, and an ellipti-spiral in a resisting me dium. Our author, aided by this correction of his error, and by 30 LIFE OF SIR ISAAC NEWTON. the discovery that a projectile would move in an elliptical orbil when under the influence of a force varying inversely as the square of the distance, was led to discover " the theorem bj which he afterwards examined the ellipsis ;" and to demonstrate the celebrated proposition that a planet acted upon by an attrac tive force varying inversely as the squares of the distances will describe an elliptical orbit, in one of whose foci the attractive force resides. When he was attending a meeting of the Royal Society, in June 1682, the conversation fell upon the subject of the measure ment of a degree of the meridian, executed by M. Picard, a French Astronomer, in 1679. Newton took a memorandum oi the result ; and afterward, at the earliest opportunity, computed from it the diameter of the earth : furnished with these new data, he resumed his calculation of 1666. As he proceeded therein, he saw that his early expectations were now likely to be realized ; the thick rushing, stupendous results overpowered him ; he be came unable to carry on the process of calculation, and intrusted its completion to one of his friends. The discoverer had, indeed, grasped the master-fact. The law of falling bodies at the earth's surface was at length identified with that which guided the moon in her orbit. And so his GREAT THOUGHT, that had for sixteen years loomed up in dim, gigantic outline, amid the first dawn of a plausible hypothesis, now stood forth, radiant and not less grand, in the mid-day light of demonstrated truth. It were difficult, nay impossible to imagine, even, the influence of a result like this upon a mind like Newton's. It was as if the keystone had been fitted to the glorious arch by which his spirit should ascend to the outskirts of infinite space — spanning the immea surable — weighing the imponderable — computing the incalculable —mapping out the marchings of the planets, and the far-wander ings of the comefs, and catching, bring back to earth some clearer notes of that higher melody which, as a sounding voice, bears perpetual witness to the design and omnipotence of a creating Deity. Newton, extending the law thus obtained, composed a series of about twelve propositions on the motion of the primary planets LIFE OF SIR ISAAC NEWTON. 31 about the sun. These were sent to London, and communicated to the Royal Society about the end of 1683. At or near this pe riod, other philosophers, as Sir Christopher Wren, Dr. Halley, and Dr. Hooke, were engaged in investigating the same subject ; but with no definite or satisfactory results. Dr. Halley, having seen, it is presumed, our author's propositions, went in August, 1684, to Cambridge to consult with him upon the subject. Newton assured him that he had brought the demonstration to perfection. In November, Dr. Halley received a copy of the work ; and, in the following month^ announced . it to the Royal Society, with the author's promise to have it entered upon their Register. Newton, subsequently reminded by the Society of his promise, proceeded in the diligent preparation of the work, and. though suffering an interruption of six weeks, transmitted the manuscript of the first book to London before the end of April. The work was entitled PHILOSOPHI/E NATURALIS PRINCIPIA MATHEMATICA, dedicated to the Royal Society, and presented thereto on the 28th of April, 1685-6. The highest encomiums were passed upon it ; and the council resolved, on the 19th of May, to print it at the expense of the Society, and under the di rection of Dr. Halley. The latter, a few days afterward, com municated these steps to Newton, who, in a reply, dated the 20th of June, holds the following language : — " The proof you sent me I like very well. I designed the whole to consist of three books ; the second was finished last summer, being short, and only wants transcribing, and drawing the cuts fairly. Some new propositions I have since thought on, which I can as well let alone. The third wants the theory of comets. In autumn last, I spent two months in calculation to no purpose for want of a good method, which made me afterward return to the first book, and enlarge it with diverse propositions, some* relating to comets, others to other things found ouf last winter. The third I now design to sup press. Philosophy is such an impertinently litigious lady, that a man had as good be engaged in l«iw-suits as have to do with her. I found it so formerly, and now I can no sooner come near her again, but she gives me warning. The first two books without the third will not so well bear the title of P/iilosophicc Naturalis 3 32 LIFE OF SIR ISAAC NEWTON. Principia Mathematicia ; and thereupon I had altered it to this, De Motu Corporum Libri duo. But after second thought I re tain the former title. It will help the sale of the book, which I ought not to diminish now 'tis yours." This " warning" arose from some pretensions put forth by Dr. Hooke. And though Newton gave a minute and positive refuta tions of such claims, yet, to reconcile all differences, he gener ously added to Prop. IV. Cor. 6, Book I, a Scholium, in which Wren, Hooke and Halley are acknowledged to have indepen dently deduced the law of gravity from the second law of Kepler. The suppression of the third book Dr. Halley could not endure to see. " I must again beg you" says he, " not to let your re sentments run so high as to deprive us of your third book, where in your applications of your mathematical doctrine to the theory of comets, and several curious experiments, which, as I guess by what you write ought to compose it, will undoubtedly render it acceptable to those who will call themselves philosophers without mathematics, which are much the greater number." To these solicitations Newton yielded. There were no "resentments," how ever, as we conceive, in his " design to suppress." He sought peace ; for he loved and valued it above all applause. But, in spite of his efforts for tranquillity's sake, his course of discovery was all along molested by ignorance or presumptuous rivalry. The publication of the great work now went rapidly forwards, The second book was sent to the Society, and presented on the 2d March ; the third, on the 6th April ; and the whole was com pleted and published in the month of May, 1686-7. In the sec ond Lemma of the second book, the fundamental principle of his fiuxionary calculus was, for the first time, given to the world ; but its algorithm or notation did not appear till published in the second volume nf Dr. Wallis's works, in 1693. And thus was ushered into existence The PRINCIPIA — a work to which pre-eminence above all the productions of the human intellect has been awarded — a work that must be esteemed of priceless worth so long as Science has a votary, or a single wor shipper be left to kneel at the altar of Truth. LIFE OF SIR ISAAC NEWTON. 33 The entire work bears the general title of THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY. It consists of three books: the first two, entitled, OF THE MOTION OF BODIES, are occupied with the laws and conditions of motions and forces, and are illus trated with many scholia treating of some of the most general and best established points in philosophy, such as the density and resistance of bodies, spaces void of matter, and the motion of sound and light. From these principles, there is deduced, in the third book, drawn up in as popular a style as possible and entitled, OF THE SYSTEM OF THE WORLD, the constitution of the system of i he world. In regard to this book, the author say^ — " I had, indeed, composed the third Book in a popular method, that it might be read by many ; but afterwards, considering that such as had not suf- ficently entered into the principles could not easily discover the strength of the consequences, nor lay aside the prejudices to which they had been many years accustomed, therefore, to prevent dis putes which might be raised upon such accounts, I chose to reduce the substance of this Book into the form of Propositions (in the mathematical way), which should be read by those only who had first made themselves masters of the principles established in the preceding Books : not that I would advise any one to the previous study of every Proposition of those Books." — "It is enough it one carefully reads the Definitions, the Laws of Motion, and the three first Sections of the first Book. He may then pass on to this Book, and consult such of the remaining Propositions of the first two Books, as the references in this, and his occasions shall re quire." So that " The System of the World" is composed both " in a popular method," and in the form of mathematical Propo sitions. The principle of Universal Gravi' ition, namely, that every particle of matter is attracted by, or gravitates to, every other particle of matter, icith a force inversely proportional to the squares of their distances — is the discovery w? ich characterizes The PRINCIPIA. This principle the author deduced from the mo tion of the moon, and the three laws of Kepler — laws, which Newton, in turn, by his greater law, demonstrated to be true. From the first law of Kepler, namely, the proportionality of LIFE OF SIR ISAAC NEWTON. the areas to t\ie times of their description, our author inferred that the force which retained the planet in its orbit was always directed to the sun ; and from the second, namely, that every planet moves in an ellipse with the sun in one of its foci, he drew the more general inference that the force by which the planet moves round that focus varies inversely as the square of its dis tance therefrom : and he demonstrated that a planet acted upon by such a force could not move in any other curve than a conic section ; showing when the moving body would describe a circu lar, an elliptical, a parabolic, or hyperbolic orbit. He demon strated, too, that this force, or attracting, gravitating power re sided in every, the least particle ; but that, in spherical masses, it operated as if confined to their centres ; so that, one sphere or body will act upon another sphere or body, with a force directly proportional to the quantity of matter, and inversely as the square of the distance between their centres; and that their velocities of mutual approach will be in the inverse ratio of their quantities o*' matter. Thus he grandly outlined the Universal Law. Verify ing its truth by the motions of terrestrial bodies, then by those of the moon and other secondary orbs, he finally embraced, in one mighty generalization, the entire Solar System — all the move ments of all its bodies — planets, satellites and comets — explain ing and harmonizing the many diverse and theretofore inexplica ble phenomena. Guided by the genius of Newton, we see sphere bound to sphere, body to body, particle to particle, atom to mass, the min utest part to the stupendous whole — each to each, each to all, and all to each — in the mysterious bonds of a ceaseless, recipro cal influence. An influence whose workings are shown to be alike present in the globular dew-drop, or oblate-spheroidal earth ; in the falling shower, or vast heaving ocean tides ; in the flying thistle-down, or fixed, ponderous rock ; in the swinging pendulum, or time-measuring sun ; in the varying and unequal moon, or earth's slowly retrograding poles ; in the uncertain meteor, or oiazing comet wheeling swiftly away on its remote, yet determined round. An influence, in fine, that may link system to system through all the star-glowing firmament ; then firmament to iirma- LIFE OF SIR ISAAC NEWTON. 35 merit ; aye, firmament to firmament, again and again, till, con verging home, it may be, to some ineffable centre, where more presently dwells He who inhabiteth immensity, and where infini tudes meet and eternities have their condux, and where around move, in softest, swiftest measure, all the countless hosts that crowd heaven's fathomless deeps. And yet Newton, amid the loveliness and magnitude of Om nipotence, lost not sight of the Almighty One. A secondary, however universal, was not taken for the First Cause. An im pressed force, however diffused and powerful, assumed not the functions of the creating, giving Energy. Material beauties, splendours, and sublimities, however rich in glory, and endless in extent, concealed not the attributes of an intelligent Supreme. From the depths of his own soul, through reason and the WORD, he had risen, a priori, to God : from the heights of Omnipotence, through the design and law of the builded universe, he proved </ posteriori, a Deity. " I had," says he, " an eye upon such prin ciples as might work, with considering men, for the belief of a Deity," in writing the PRINCIPIA ; at the conclusion whereof, he teaches that — " this most beautiful system of the sun, planets and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other like systems, these, being forme 1 by the like wise counsels, must be all subject to the dominion of One ; especially since the light of the fixed stars is of the same nature with the light of the sun, and from every system light passes into all other systems : and lest the systems of the fixed stars should, by their gravity, fall on each other mutually, he hath placed those systems at immense distances one from another. " This Being governs all things, not as the soul of the world, but as Lord over all ; and on account of his dominion he is wont, to be called Lord God Travrowparwp or Universal Ruler ; for God is a relative word, and has a respect to servants ; and Deity is the dominion of God, not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect ; but a being, however perfect, without dominion, cannot be said to 36 LIFE OF SIR ISAAC NEWTON. be Lord God ; for we say, my God, your God, the God of Israel the God of Gods, and Lord of Lords ; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods : we do not say my Infinite, or my Perfect : these are titles which have no respect to servants. The word God usually signifies Lord ; but every Lord is not God. It is the dominion of a spir itual Being which constitutes a God ; a true, supreme, or imagi nary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent and powerful Being ; and from his other perfections, that he is supreme or most perfect. He is eternal and in finite, omnipotent and omniscient ; that is, his duration reaches from eternity to eternity ; his presence from infinity to infinity ; he governs all things and knows all things, that are or can be done. He is not eternity or infinity, but eternal and infinite ; he is not duration and space, but he endures and is present. He endures forever and is everywhere present ; and by existing always and everywhere, he constitutes duration and space. Since every particle of space is always, and every indivisible moment of duration is everywhere, certainly the Maker and Lord of things cannot be never and nowhere. Every soul that has perception is, though in different times and different organs of sense and mo tion, still the same indivisible person. There are given succes sive parts in duration, co-existent parts in space, but neither the one nor the other in the person of a man, or his thinking principle ; and much less can they be found in the thinking sub stance of God. Every man. so far as he is a thing that has j:er- ceptiori, is one and the same man during his whole life, in all and each of his organs of sense. God is one and the same God, al ways and everywhere. He is omnipresent, not virtually only, but also substantially ; for virtue cannot subsist without sub stance. In him are all things contained and moved ; yet neither affects the other ; God suffers nothing from the motion of bodies ; bodies find no resistance from the omnipresence of God. It is allowed by all that the Supreme God exists necessarily ; and by the same necessity he exists always and everywhere. Whence also he is all similar, all eye, all ear, all brain, all arm, all powei LIFE CF SIR ISAAC NEWTON. 37 to perceive, to understand, and to act ; but in a manner not at all human, in a manner not at all corporeal, in a manner utterly un known to us. As a blind man has no idea of colours, so have we no idea of the manner by which the all-wise God perceives and understands all things. He is utterly void of all body, and bodily figure, and can therefore neither be seen, nor heard, nor touched ; nor ought he to be worshipped under the representation of any corporeal thing. We have ideas of his attributes, but what the real substance of anything is we know not. In bodies we see only their figures and colours, we hear only the sounds, we touch only their outward surfaces, we smell only the smells, and taste only the savours ; but their inward substances are not to be known, either by our senses, or by any reflex act of our minds : much less, then, have we any idea of the substance of God. We know him only by his most wise and excellent contrivances of things, and final causes ; we admire him for his perfections ; but we rev erence and adore him on account of his dominion ; for we adore him as his servants ; and a god without dominion, providence, and final causes, is nothing else but Fate and Nature. Blind meta physical necessity, which is certainly the same always and every where, could produce no variety of things. All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being neces sarily existing." Thus, the diligent student of science, the earnest seeker of truth, led, as through the courts of a sacred Temple, wherein, at each step, new wonders meet the eye, till, as a crowning grace, they stand before a Holy of Holies, and learn that all science and all truth are one which hath its beginning and its end in the knowledge of Him whose glory the heavens declare, and whose handiwork the firmament showeth forth. The introduction of the pure and lofty doctrines of the PRIN- CIPIA was perseveringly resisted. Descartes, with his system of vortices, had sown plausibly to the imagination, and error had struck down deeply, and shot up luxuriantly, not only in the popular, but in the scientific mind. Besides the idea — in itself so simple and so grand — that the great masses of the planets were 38 LIFE OF SIR ISAAC NEWTON. suspended in empty space, and retained in their orbits by an in visible influence residing in the sun — was to the ignorant a thing inconceivable, and to the learned a revival of the occult qualities of the ancient physics. This remark applies particularly to the continent. Leibnitz misapprehended ; Huygens in part rejected ; John Bernouilli opposed ; and Fontenelle never received the doc trines of the PRINCIPIA. So that, the saying of Voltaire is prob ably true, that though Newton survived the publication of his great work more than forty years, yet, at the time of his death, lie had not above twenty followers out of England. But in England, the reception of our author's philosophy was rapid and triumphant. His own labours, while Lucasian Pro fessor ; those of his successors in that Chair — Whiston and Saunderson ; those of Dr. Samuel Clarke, Dr. Laughton, Roger Cotes, and Dr. Bentley ; the experimental lectures of Dr. Keill and Desaguliers ; the early and powerful exertions of David Gregory at Edinburgh, and of his brother James Gregory at St. Andrew's, tended to diffuse widely in England and Scotland a knowledge of, and taste for the truths of the PRINCIPIA. Indeed, its mathematical doctrines constituted, from the first, a regular part of academical instruction ; while its physical truths, given to the public in popular lectures, illustrated by experiments, had, before the lapse of twenty ) ( ar.s, become familiar to, and adopted by the general mind. Pemberton's popular " View of Sir Isaac Newton's Philosophy" was published, in 1728 ; and the year after ward, an English translation of the PRINCIPIA, and System of the World, by Andrew Motte. And since that period, the labours of Le Seur and Jacquier, of Thorpe, of Jebb, of Wright and others have greatly contributed to display the most hidden treasures of the PRINCIPIA. About the time of the publication of the Principia, James II., bent on re-establishing the Romish Faith, had, among other ille gal acts, ordered by mandamus, the University of Cambridge to confer the degree of Master of Arts upon an ignorant monk. Obedience to this mandate was resolutely refused. Newton was one of the nine delegates chosen to defend the independence of the University. They appeared before the High Court ; — and LIFE OF SIR ISAAC NEWTON. 39 successfully : the king abandoned his design. The prominent part which our author took in these proceedings, and his eminence in the scientific world, induced his proposal as one of the parlia mentary representatives of the University. He was elected, in 1688, and sat in the Convention Parliament till its dissolution. After the first year, however, he seems to have given little or no attention to his parliamentary duties, being seldom absent from the University till his appointment in the Mint, in 1695. Newton began his theological researches sometime previous to 1691 ; in the prime of his years, and in the matured vigour of his intellectual powers. From his youth, as we have seen, he had devoted himself with an activity the most unceasing, and an energy almost superhuman to the discovery of physical truth ; — giving to Philosophy a new foundation, and to Science a new temple. To pass on, then, from the consideration of the material, more directly to that of the spiritual, was a natural, nay, with so large and devout a soul, a necessary advance. The Bible was to him of inestimable worth. In the elastic freedom, which a pure and unswerving faith in Him of Nazareth gives, his mighty facul ties enjoyed the only completest scope for development. His original endowment, however great, combined with a studious application, however profound, would never, without this libera tion from the dominion of passion and sense, have enabled him to attain to that wondrous concentration and grasp of intellect, for which Fame has as yet assigned him no equal. Gratefully he owned, therefore, the'same Author in the Book of Nature and the Book of Revelation. These were to him as drops of the same unfathomable ocean ; — as outrayings of the same inner splendour ; — as tones of the same ineffable voice ; — as segments of the same infinite curve. "With great joy he had found himself enabled to proclaim, as an interpreter, from the hieroglyphs of Creation, the existence of a God : and now, with greater joy, and in the fulness of his knowledge, and in the fulness of his strength, he laboured to make clear, from the utterances of the inspired Word, the far mightier confirmations of a Supreme Good, in all its glorious amplitude of Being and of Attribute ; and to bring the infallible workings thereof plainly home to the understandings and the 40 LIFE OF SIR ISAAC NEWTON. •affections of his fellow-men ; and finally to add the weight of his own testimony in favour of that Religion, whose truth is now. in deed, " girded with the iron and the rock of a ponderous and co lossal demonstration." His work, entitled, OBSERVATIONS UPON THE PROPHECIES OF HOLY WRIT, PARTICULARLY THE PROPHECIES OF DANIEL AND THE APOCALYPSE OF ST. JOHN, first published in London, in 1733 4to. consists of two parts : the one devoted to the Prophecies oi Daniel, and the other to the Apocalypse of St. John. In the first part, he treats concerning the compilers of the books of the Old Testament ; — of the prophetic language ; — of the vision of the four beasts ; — of the kingdoms represented by the feet of the image composed of iron and clay ; — of the ten kingdoms repre sented by the ten horns of the beast ; — of the eleventh horn of Daniel's fourth beast ; of the power which should change times and laws ;— of the kingdoms represented in Daniel by the ram and he-goat ; — of the prophecy of the seventy weeks ; — of the times of the birth and passion of Christ ; — of the prophecy of the Scripture of Truth ; — of the king who doeth according to his will, and magnified himself above every god, and honoured Mahuzzims, and regarded not the desire of women ; — of the Mahuzzim, hon oured by the king who doeth according to his will. In the sec ond part, he treats of the time when the Apocalypse was written , of the scene of the vision, and the relation which the Apocalypse has to the book of the law of Moses, and to the worship of God in the temple ; — of the relation which the Apocalypse has to the prophecies of Daniel, and of the subject of the prophecy itself Newton regards the prophecies as given, not for the gratification of man's curiosity, by enabling him to foreknow ; but for his con viction that the world is governed by Providence, by witnessing their fulfilment. Enough of prophecy, he thinks, has already been fulfilled to afford the diligent seeker abundant evidence of God's providence. The whole work is marked by profound erudition, sagacity and argument. And not less learning, penetration and masterly reasoning are conspicuous in his HISTORICAL ACCOUNT OF Two NOTABLE CORRUPTIONS OF SCRIPTURES IN A LETTER TO A FRIEND. This LIFE OF SIR ISAAC NEWTON. 41 Treatise, first accurately published in Dr. Horsley's edition of his works, relates to two texts : the one, 1 Epistle of St. John v. 7 ; the other, 1 Epistle of St. Paul to Timothy iii. 16. As this work had the effect to deprive the advocates of the doctrine of the Trinity of two leading texts, Newton has been looked upon as an Arian ; but there is absolutely nothing in his writings to warrant such a conclusion. His regaining theological works consist of the LEXICON PRO- PHETICUM, which was left incomplete ; a Latin Dissertation on the sacred cubit of the Jews, which was translated into English, and published, in 1737. among the Miscellaneous Works of John Greaves ; and FOUR LETTERS addressed to Dr. Bentlty, contain ing some arguments in proof of a Deity. These Letters were dated respectively : 10th December, 1692 ; 17th January, 1693 ; 25th February, 1693; and llth February, 1693— the fourth bearing an earlier date than the third. The best faculties and the profoundest acquirements of our author are convincingly manifest in these lucid and powerful compositions. They were published in 1756, and reviewed by Dr. Samuel Johnson. Newton's religious writings are distinguished by their absolute freedom from prejudice. Everywhere, throughout them, there glows the genuine nobleness of soul. To his whole life, indeed, we may here fitly extend the same observation. He was most richly imbued with the very spirit of the Scriptures which he so delighted to study and to meditate upon. His was a piety, so fervent, so sincere and practical, that it rose up like a holy incense from every thought and act. His a benevolence that not only willed, but endeavoured the best for all. His a philanthropy that held in the embracings of its love every brother-man. His a toleration of the largest and the truest ; condemning per secution in every, even its mildest form ; and kindly encouraging each striving after excellence : — .1 toleration that came not of indifference — for the immoral and the impious met with their quick rebuke — but a toleration that came of the wise humbleness and the Christian charity, which see, in the nothingness of self and the almightiness of TRUTH, no praise for the ablest, and no blame for th^ feeblest in their strugglings upward to light and life. 42 LIFE OF SIR ISAAC NEWTON, Tn the winter of 1691-2, on returning from chapel, one morn ing, Newton foima tnat a favourite little dog, called Diamond, had overturned a lighted taper on his desk, and that several pa pers containing the results of certain optical experiments, were nearly consumed. His only exclamation, on perceiving his loss, was, " Oh Diamond, Diamond, little knowest thou the mischiel thou hast done," Dr. Brewster, in his life of our author, gives the following extract from the manuscript Diary of Mr. Abraham De La Pryme. a student in the University at the time of this oc currence. " 1692. February, 3. — What I heard to-day I must relate. There is one Mr. Newton (whom I have very oft seen), Fellow of Trinity College, that is mighty famous for his learning, being a most excellent mathematician, philosopher, divine, &c. He has been Fellow of the Royal Society these many years ; and among other very learned books and tracts, he:s written one upon the mathe matical principles of philosophy, which has given him a mighty name, he having received, especially from Scotland, abundance of congratulatory letters for the same ; but of all the books he ever wrote, there was one of colours and light, established upon thou sands of experiments which he had been twenty years of making, and which had cost him many hundreds of pounds. This book which he vaiued so much, and which was so much talked of, had the ill luck to perish, and be utterly lost just when the learned author was almost at pitting a conclusion at the same, after this manner : In a winter's morning, leaving it among his other papers on his study table while he went to chapel, the candle, which he had unfortunately left burning there, too, catched hold by some means of other papers, and they fired the aforesaid book, and ut terly consumed it and several other valuable writings ; arid which is most wonderful did no further mischief. But when Mr. New ton came from chapel, and had seen what was done, every one thought he would have run mad, he was so troubled thereat that he was not himself for a month after. A long account of this his system of colours you may find in the Transactions of the Royal Society, which he had sent up to them long before this sad mis chance happened unto him." LIFE OF SIR ISAAC NEWTON. 43 It will be borne in mind that all of Newton's theological wri tings, with the exception of the Letters to Dr. Bentley, were composed before this event which, we must conclude, from Pryme's words, produced a serious impression upon our author for about a month. But M. Biot, in his Life of Newton, relying on a memorandum contained in a small manuscript Journal of Huygens, declares this occurrence to have caused a deran-gement of New ton's intellect. M. Blot's opinions and deductions, however, as well as those of La Place, upon this subject, were based upon erroneous data, and have been overthrown by the clearest proof. There is not, in fact, the least evidence that Newton's reason was, for a single moment, dethroned ; on the contrary, the testimony is conclusive that he was, at all times, perfectly capable of carry ing on his mathematical, metaphysical and astronomical inquiries. Loss of sleep, loss of appetite, and irritated nerves will disturb somewhat the equanimity of the most serene ; and an act done, or language employed, under such temporary discomposure, is not a just criterion of the general tone and strength of a man's mind. As to the accident itself, we may suppose, whatever might have been its precise nature, that it greatly distressed him, and, still further, that its shock may have originated the train of nervous derangements, which afflicted him, more or less, for two years afterward. Yet, during this very period of ill health, we find him putting forth his highest powers. In 1692, he prepared for, and transmitted to Dr. Wallis the first proposition of the Treatise on Quadratures, with examples of it in first, second and third flux ions. He investigated, in the same year, the subject of haloes ; making and recording numerous and important observations rela tive thereto. Those profound and beautiful Letters to Dr. Bentley were written at the close of this and the beginning of the next year. In October, 1693, Locke, who was then about publishing a second edition of his work on the Human Understanding, request ed Newton to reconsider his opinions on innate ideas. And in 1694, he was zealously occupied in perfecting his lunar theory ; visiting Flamstead, at the Royal Observatory of Greenwich, in September, and obtaining a series of lunar observations ; and 14 LIFE OF SIR ISAAC NEWTON. commencing, in October, a correspondence with that distinguished practical Astronomer, which continued till 1698. We now arrive at the period when Newton permanently with drew from the seclusion of a collegiate, and entered upon a more active and public life. He was appointed Warden of the Mint, in 1695, through the influence of Charles Montague, Chancellor of the Exchequer, and afterward Earl of Halifax. The current roin of the nation had been adulterated and debased, and Mon tague undertook a re-coinage. Our author's mathematical and chemical knowledge proved eminently useful in accomplishing this difficult and most salutary reform. In 1699, he was pro moted to the Mastership of the Mint — an office worth twelve or fifteen hundred pounds per annum, and which he held during the remainder of his life. He wrote, in this capacity, an official Re port on the Coinage, which has been published ; he also prepared a Table of Assays of Foreign Coins, which was printed at the end of Dr. Arbuthnot's Tables of Ancient Coins, Weights, and Measures, in 1727. Newton retained his Professorship at Cambridge till 1703. But he had, on receiving the appointment of Master of the Mint, in 1699, made Mr. Whiston his deputy, with all the emoluments of the office ; and, on finally resigning, procured his nomination to the vacant Chair. In January 1697, John Bernouilli proposed to the most distin guished mathematicians of Europe two problems for solution. Leibnitz, admiring the beauty of one of them, requested the time for solving it to be extended to twelve months — twice the period originally named. The delay was readily granted. Newton, how ever, sent in, the day after he received the problems, a solution of them to the President of the Royal Society. Bernouilli obtained solutions from Newton, Leibinitz and the Marquis De L'Hopital ; but Newton's though anonymous, he immediately recognised " tanquam ungue leonem" as the lion is known by his claw. We may mention here the famous problem of the trajectories proposed by Leibnitz, in 1716, for the purpose of "feeling the pulse of the English Analysts." Newton received the problem about five o'clock in the afternoon, as he was returning from the LIFE OF SIR ISAAC NEWTON. 45 Mint ; and though it was extremely difficult and he himself much fatigued, yet he completed its solution, the same evening before he went to bed. The history of these problems affords, by direct comparison, a striking illustration of Newton's vast superiority of mind. That amazing concentration and grasp of intellect, of which we have spoken, enabled him to master speedily, and, as it were, by a single effort, those things, for the achievement of which, the many would essay utterly in vain, and the very, very few attain only after long and renewed striving. And yet, with a modesty as unparalleled as his power, he attributed his successes, not to any extraordinary sagacity, but solely to industry and patient thought. Mr- kept the subject of consideration constantly before him, and waited till the first dawning opened gradually into a full and clear light ; never quitting, if possible, the mental process till the object of it were wholly gained. He never allowed this habit of meditation to appear in his intercourse with society ; but in the privacy of his own chamber, or in the midst of his own family, he gave himself up to the deepest abstraction. Occupied with some interesting investigation, he would often sit down on his bedside, after he rose, and remain there, for hours, partially dressed. Meal-time would frequently come and pass unheeded ; so that, unless urgently reminded, he would neglect to take the re quisite quantity of nourishment. But notwithstanding his anx iety to be left undisturbed, he would, when occasion required, turn aside his thoughts, though bent upon the most intricate re search, and then, when leisure served, again direct them to the very point where they ceased to act : and this he seemed to ac complish not so much by the force of his memory, as by the force of his inventive faculty, before the vigorous intensity of which, no subject, however abstruse, remained long unexplored. Me was elected a member of the Royal Academy of Sciences at Paris, in 1699, when that distinguished Body were empowered, by a new charter, to admit a small number of foreign associates. In 1700, he communicated to Dr. Halley a description of his re flecting instrument for observing the moon's distance from the fixed stars. This description was published in the Philosophical 46 LIFE OF SIR ISAAC NEWTON, Transactions, in 1742. The instrument was the same as that produced by Mr. Hadley, in 1731, and which, under the name of Hadley's Quadrant, has been of so great use in navigation. On the assembling of the new Parliament, in 1701, Newton was re- elected one of the members for the University of Cambridge. In 1703, he was chosen President of the Royal Society of London, to which office he was annually re-elected till the period of his decease — about twenty-five years afterward. Our author unquestionably devoted more labour to, and, in many respects, took a greater pride in his Optical, than his other discoveries. This science he had placed on a new and indestruc tible basis ; and he wished not only to build, but to perfect the costly and glowing structure. He had communicated, before the publication of the PRINCIPIA, his most important researches on light to the Royal Society, in detached papers which were inserted in successive numbers of the Transactions ; but he did not pub lish a connected view of these labours till 1704, when they appeared under the title of OPTICS : OR, A TREATISE ON THE REFLEXIONS, REFRACTIONS, INFLEXIONS AND COLOURS OF LIGHT. To this, but to no subsequent edition, were added two Mathematical Trea tises, entitled, TRACTATUS DUO DE SPECIEBUS ET MAGNITUDINE FIGURARUM cuRViLiNEARUM ; the one bearing the title TRACTATUS DE QUADRATURA CuRVARUM ; and the other, that of ENUMERATIO LINEARUM TERTII ORDiNis. The publication of these Mathemati cal Treatises was made necessary in consequence of plagiarisms from the manuscripts of them loaned by the author to his friends. Dr. Samuel Clarke published a Latin translation of the Optics, in in 1706 ; whereupon he was presented by Newton, as a mark of his grateful approbation, with five hundred pounds, or one hun dred pounds for each of his children. The work was afterward translated into French. It had a remarkably wide circulation, and appeared, in several successive editions, both in England and on the Continent. There is displayed, particularly on this Opti cal Treatise, the author's talent for simplifying and communica ting the profoundest speculations. It is a faculty rarely united to that of the highest invention. Newton possessed both ; and thus that mental perfectness which enabled him to create, to combine, LIFE OF SIR ISAAC NEWTON. 47 and to teach, and so render himself, not the "ornament" cnly; but inconceivably more, the pre-eminent benefactor of his species. The honour of knighthood v/as conferred on our author in 1705. Soon afterward, he was a candidate again for the Repre sentation of the University, but was defeated by a large majority. It is thought that a more pliant man was preferred by both min isters and electors. Newton was always remarkable for simplicity of dress, and his only known departure from it was on this oc casion, when he is said to have appeared in a suit of laced clothes. The Algebraical Lectures which he had, Juring nine years, delivered at Cambridge, were published by Whiston, in 1707, under the title of ARITHMETICS UNIVERSALIS, SINE DE COMPOSI TIONS ET RESOLUTIONS ARITHMETICA LIBER. This publication is said to have been a breach of confidence on Whiston's part. Mr. Ralphson, not long afterward, translated the work into English ; and a second edition of it, with improvements by the author, was issued at London, 1712, by Dr. Machin. Subsequent editions, both in English and Latin, with commentaries, have been published. In June, 1709, Newton intrusted the superintendence of a second edition of the PRINCIPIA to Roger Cotes, Plumian Pro fessor of Astronomy at Cambridge. The first edition had been sold off for some time. Copies of the work had become very rare, and could only be obtained at several times their original cost. A great number of letters passed oetween the author and Mr. Cotes during the preparation of the edition, which finally appeared in May, 1713. It had many alterations and improve ments, and was accompanied by an admirable Preface from the pen of Cotes. Our author's early Treatise, entitled, ANALYSIS PER EQUATIONES NUMERO TERMINORUM INFINITAS, as well as a small Tract, Gearing the title of METHODUS DIFFERENTIALS, was published, witn nis consent, in 1711. The' former of these, and the Treatise De Quadratura Curvarum, translated into Englisn, witn a .arge com mentary, appeared in 1745. His work, entitled. ARTIS ANA LYTICS SPECIMINA, VEL GEOMETRIA ANALYTICA, was iirs; given to the world in the edition of Dr. Horsley, 1779. 48 LIFE OF SIR ISAAC NEWTON. It is a notable fact, in Newton's history, that he never volun* tarily published any one of his purely mathematical writings The cause of this unwillingness in some, and, in other instances, of his indifference, or, at least, want of solicitude to put forth his works may be confidently sought for in his repugnance to every thing like contest or dispute. But, going deeper than this aver sion, we find, underlying his whole character and running parallel with all his discoveries, that extraordinary humility which always preserved him in a position so relatively just to the behests of time and eternity, that the infinite value of truth, and the utter worthlessness of fame, were alike constantly present to him. Judging of his course, however, in its more temporary aspect, as bearing upon his immediate quiet, it seemed the most unfortunate. For an early publication, especially in the case of his Method of Fluxions, would have anticipated all rivalry, and secured him from the contentious claims of Leibnitz. Still each one will solve the problem of his existence in his own way, and, with a manlike Newton, his own, as we conceive, could be no other than the best way. The conduct of Leibnitz in this affair is quite irreconcilable with the stature and strength of the man ; giant-like, and doing nobly, in many ways, a giant's work, yet cringing himself into the dimensions and performances of a common calumniator. Opening in 1699, the discussion in question continued till the close of Leibnitz's life, in 1716. We give the summary of the case as contained in the Report of the Committee of the Royal Society, the deliberately weighed opinion of which has been adopted as an authoritative decision in all countries. " We have consulted the letters and letter books in the custody of the Royal Society, and those found among the papers of Mr. John Collins, dated between the years 1669 and 1677, inclusive ; and showed them to such as knew and avouched the hands of Mr. Barrow, Mr. Collins, Mr. Oldenburg, and Mr. Leibnitz ; and compared those of Mr. Gregory with one another, and with copies of some of them taken in the hand of Mr. Collins ; and have extracted from them what relates to the matter referred to us : all which extracts, herewith delivered to you, we believe to be genuine and authentic. And by these letters and papers wf find: — LIFE OF SIR ISAAC NEWTON. 49 " I. Mr. Leibnitz was in London in the beginning of the year 1673 ; and went thence in or about March, to Paris, where he kept a correspondence with Mr. Collins, by means of Mr. Olden burg, till about September, 1676, and then returned, by London and Amsterdam, to Hanover: and that Mr. Collins was very free in communicating to able mathematicians what he had received from Mr, Newton and Mr. Gregory. " II. That when Mr. Leibnitz was the first time in London, he contended for the invention of another differential method, properly so called ; and, notwithstanding he was shown by Dr. Pell that it was Newton?s method, persisted in maintaining it to be his own invention, by reason that he had found it by himself without knowing what Newton had done before, and had much improved it. And we find no mention of his having any other differential method than Newton's before his letter of the 21st of June, 1677, which was a year after a copy of Mr. Newton's letter of the 10th of December, 1672, had been sent to Paris to be communicated to him ; and above four years after Mr. Collins began to communicate that letter to his correspondents ; in which letter the method of fluxions was sufficiently described to any intelligent person. "III. That by Mr. Newton's letter, of the 13th of June, 1676 it appears that he had the method of fluxions above five years before the writing of that letter. And by his Analysis per ^Equa- tiones numero Terminorum Infmitas, communicated by Dr. Barrow to Mr. Collins, in July, 1669, we find that he had invented the method before that time. "IV. That the differential method is one and the same with the method of fluxions, excepting the name and mode of notation ; Mr. Leibnitz calling those quantities differences which Mr. Newton calls moments, or fluxions ; and marking them with a letter d — a mark not used by Mr. Newton. " And, therefore, we take the proper question to be, not who invented this or that method, but, who was the first inventor of the method ? And we believe that those who have reputed Mr. Leibnitz the first inventor knew little or nothing of his correspond ence with Mr. Collins and Mr. Oldenburg long before, nor of Mr. 50 LIFE OP SIR ISAAC NEWTON. Newton's hiving that method above fifteen years before Mr Leibnitz began to publish it in the Acta Eruditorum of Leipsic. " For which reason we reckon Mr. Newton the first inventor ; and are of opinion that Mr. Keill, in asserting the same, has been no ways injurious to Mr. Leibnitz. And we submit to the judg ment of the Society, whether the extract and papers, now pre sented to you, together with what is extant, to the same pur pose, in Dr. Wallis's third volume, may not deserve to be made public." This Report, with the collection of letters and manuscripts, under the title of COMMERCIUM EPISTOLICUM D. JOHANNIS COLLINS ET ALIORUM DE ANALYSI PROMOTA JuSSU SoCIETATIS REGIES EDITUM, appeared accordingly in the early part of 1713. Its publication seemed to infuse additional bitterness into the feelings of Leibnitz, who descended to unfounded charges and empty threats. He had been privy counsellor to the Elector of Han over, before that prince was elevated to the British throne ; and in his correspondence, in 1715 and 1716, with the Abbe Conti, then at the court of George L, and with Caroline, Princess of Wales, he attacked the doctrines of the PRINCIPIA, and indirectly its author, in a manner very discreditable to himself, both as a learned and as an honourable man. His assaults, however, were triumphantly met; and, to the complete overthrow of his rival pretensions, Newton was induced to give the finishing blow. The verdict is universal and irreversible that the English preceded the German philosopher, by at least ten years, in the invention of fluxions. Newton could not have borrowed from Leibnitz ; but Leibnitz might have borrowed from Newton. A new edition of the Commercium Epistolicum was published in 1722-5 (?) ; but neither in this, nor in the former edition, did our author take any part. The disciples, enthusiastic, capable and ready, effectually shielded, with the buckler of Truth, the character of the Master, whose own conduct throughout was replete with delicacy, dignity and justice. He kept aloof from the controversy — in which Dr. Keill stood forth as the chief representative of the Newtonian side — till the very last, when, for the satisfaction of the King, George L. rather than for his own, he consented to put forth his LIFE OF SI| L^.-vJ NEWTON. 5i hand and firmly secure his rights upon a certain and impregnable, basis. A petition to have inventions for promoting the discovery of the longitude at sea, suitably rewarded, was presented to the House of Commons, in 1714. A committee, having been appointed to investigate the subject, called upon Newton and others for their opinions. That of our author was given in writing, A report, favourable to the desired measure, was then taken up, and a bill for its adoption subsequently passed. On the ascension of George I., in 1714, Newton became an object of profound interest at court. His position under govern ment, his surpassing fame, his spotless character, and. above all, his deep and consistent piety, attracted the reverent regard of the Princess of Wales, afterward queen -consort to George II. She was a woman of a highly cultivated mind, and derived the greatest pleasure from conversing with Newton and corresponding with Leibnitz. One day, in conversation with her, our author men tioned and explained a new system of chronology, which he had composed at Cambridge, where he had been in the habit " of refreshing himself with history and chronology, when he wac weary with other studies." Subsequently, in the year 1718, she requested a copy of this interesting and ingenious work Newton, accordingly, drew up an abstract of the system from the separate papers in which it existed, and gave it to her on condition that it should riot be communicated to any other person. Sometime afterward she requested that the Abbe Conti might be allowed to have a copy of it The author consented: and the abbe received a copy of the manuscript, under the like injunction and promise of secrecy. This manuscript bore the title of " A short Chronicle, from the First Memory of Tilings in Europe, to the Conquest of Persia, by Alexander the Great." After Newton took up his residence in London, he lived in a style suited to his elevated position and rank. He kept his car riage, with an establishment of three male and three female serv ants. But to everything like vain show and luxury he was utterly averse. His household affairs, for the last twenty years of his life, were under the charge of his niece, Mrs. Catherine Barton, 52 LIFE OF SIR ISAAC NEWTON. wife and widow of Colonel Barton — a woman of great beauty and accomplishment — and subsequently married to John Conduit, Esq. At home Newton was distinguished by that dignified and gentle hospitality which springs alone from true nobleness. On all pro per occasions, he gave splendid entertainments, though without ostentation. In society, whether of the palace or the cottage, his manner was self-possessed and urbane ; his look benign and affable ; his speech candid and modest ; his whole air undisturb edly serene. He had none of what are usually called the singu larities of genius ; suiting himself easily to every company — except that of the vicious and wicked ; and speaking of himself and others, naturally, so as never even to be suspected of vanity. There was in him, if we may be allowed the expression, a WHOLE NESS of nature, which did not admit of such imperfections and weakness — the circle was too perfect, the law too constant, and the disturbing forces too slight to suffer scarcely any of those eccentricities which so interrupt and mar the movements of many bright spirits, rendering their course through the world more like that of the blazing meteor than that of the light and life-impart ing sun. In brief, the words GREATNESS and GOODNESS could not, humanly speaking, be more fitly employed than when applied as the pre-eminent characteristics of this pure, meek and vene rable sage. In the eightieth year of his age, Newton was seized with symptoms of stone in the bladder. His disease was pronounced incurable. He succeeded, however, by means of a strict regimen, and other precautions, in alleviating his complaint, and procuring long intervals of ease. His diet, always frugai, was now extremely temperate, consisting chiefly of broth, vegetables, and fruit, with, now and then, a little butcher meat. He gave up the use of his carriage, and employed, in its stead, when he went out, a chair. All invitations to dinner were declined ; and only small parties were received, occasionally, at his own house. In 1724 he wrote to the Lord Provost of Edinburgh, offering to contribute twenty pounds yearly toward the salary of Mr. Maclaurin, provided he accepted the assistant Professorship of Mathematics in the University of that place. Not only in the LIFE OP SIR ISAAC NEWTON. 53 cause of ingenuity and learning, but in that of religion — in relieving the poor and .assisting his relations, Newton annually expended large sums. He was generous and charitable almost to a fault. Those, he would often remark, who gave away nothing till they died, never gave at all. His wealth had become considerable by a prudent economy ; but he regarded money in no other light than as one of the means wherewith he had been intrusted to do good, and he faithfully employed it accordingly. He experienced, in spite of all his precautionary measures, a return of his complaint in the month of August, of the same year, 1 724, when he passed a stone the size of pea ; it came from him in two pieces, the one at the distance of two day.s from the other. Tolerable good health then followed for some months. In Janu ary, 1725, however, he was taken with a violent cough and inflam mation of the lungs. In consequence of this attack, he was pre vailed upon to remove to Kensington, where his health greatly improved. In February following, he was attacked in both feet with the gout, of the approach of which he had received, a few years before, a slight warning, and the presence of which now produced a very beneficial change in his general health. Mr. Conduit, his nephew, has recorded a curious conversation which took place, at or near this time, between himself and Sir Isaac. "I was, on Sunday night, the 7th March, 1724-5, at Kensing ton, with Sir Isaac Newton, in his lodgings, just after he was out of a fit of the gout, which he had had in both of his feet, for the first time, in the eighty-third year of his age. He was better after it, and his head clearer and memory stronger than I had known them for some time. He then repeated to me, by way of dis course, very distinctly, though rather in answer to my queries, than in one continued narration, what he had often hinted to me before, viz. : that it was his conjecture (he would affirm nothing) that there was a sort of revolution in the heavenly bodies ; that the vapours and light, emitted by the sun, which had their sedi ment, as water and other matter, had gathered themselves, by degrees, into a body, and attracted more matter from the planets, and at last made a secondary planet (viz. : one of those that go round another planet), and then, by gathering to them, and 54 LIFE OF SIR ISAAC NEWTON. attracting more matter, became a primary planet ; and then, bf increasing still, became a comet, which, after certain revolutions, by coming nearer and nearer to the sun, had all its volatile parts condensed, and became a matter tit to recruit and replenish the sun (which must waste by the constant heat and light it emitted), as a faggot would this fire if put into it (we were sitting by a wood fire), and that that would probably be the effect of the comet of 1680, sooner or later ; for, by the observations made upon it, it appeared, before it came near the sun, with a tail only two or three degrees long ; but, by the heat it contracted, in going so near the sun, it seemed to have a tail of thirty or forty degrees when it went frpm it ; that he could not say when this comet would drop into the sun ; it might perhaps have five or six revo lutions more first, but whenever it did it would so much increase the heat of the sun that this earth would be burned, and no ani mals in it could live. That he took the three phenomena, seen by Hipparchus, Tycho Brahe, and Kepler's disciples, to have been of this kind, for he could not otherwise account for an extraor dinary light, as those were, appearing, all at once, among the the fixed stars (all which he took to be suns, enlightening other planets, as our sun does ours), as big as Mercury or Venus seems to us, and gradually diminishing, for sixteen months, and then sinking into nothing. He seemed to doubt whether there were not intelligent beings, superior to us, who superintended these revolutions of the heavenly bodies, by the direction of the Supreme Being. He appeared also to be very clearly of opinion that the inhabitants of this world were of short date, and alledged, as one reason for that opinion, that all arts, as letters, ships, printing, needle, &c., were discovered within the memory of history, which could not have happened if the world had been eternal ; and that there were visible marks of ruin upon it which could not be effected by flood only. When I asked him how this earth could have been repeopled if ever it had undergone the same fate it was threatened with hereafter, by the comet of 1680, he answered, that required the power of a Creator. He said he took all the planets to be composed of the same matter with this earth, viz. : earth, water, stones, &c.3 but variously concocted. J LIFE OP SIR ISAAC NEWTON. 55 asked him why he would not publish his conjectures, as conjec tures, and instanced that Kepler had communicated his ; and though he had not gone near so far as Kepler, yet Kepler's guesses were so just and happy that they had been proved and demonstrated by him. His answer was, " I do not deal in con jectures." But, on my talking to him about the four observations that had been made of the comet of 1680, at 574 years' distance, and asking him the particular times, he opened his Principia, which laid on the table, and showed me the particular periods, viz.: 1st. The Julium Sidus, in the time of Justinian, in 1106, in 1680. " And I, observing that he said there of that comet, ' incidet in corpus solis,' and in the next paragraph adds, ' stellae fixae refici possunt,' told him I thought he owned there what we had been talking about, viz. : that the comet would drop into the sun, and that fixed stars were recruited and replenished by comets when they dropped into them ; and, consequently, that the sun would be recruited too ; and asked him why he would not own as fully what he thought of the sun as well as what he thought of the fixed stars. He said, 'that concerned us more;' and, laugh ing, added, that he had said enough for people to know his meaning." In the summer of 1725, a French translation of the chronolo gical MS., of which the Abbe Conti had been permitted, some time previous, to have a copy, was published at Paris, in violation of all good faith. The Punic Abbe had continued true to his promise of secrecy while he remained in England ; but no sooner did he reach Paris than he placed the manuscript into the hands of M. Freret, a learned antiquarian, who translated the work, and accompanied it with an attempted refutation of the leading points of the system. In November, of the same year, Newton received a presentation copy of this publication, which bore the title of ABREGE DE CHRONOLOGIE DE M. LE CHEVALIER NEWTON, FAIT PAR LUI-MEME, ET TRADUIT SUR LE MANUSCRIPT ANGLAIS. Soon afterward a paper entitled, REMARKS ON TFE OBERVATIONS MADE ON A CHRONOLOGICAL INDEX OF SIR ISAAC NE.WTON, TRANSLATED INTO FRENCH BY THE OBSERVATOR, ANL PUBLISHED AT PARIS, 56 LIFE OF SIR ISAAC NEWTON, was drawn up by our author, and printed in the Philosophical Transactions for 1725. It contained a history of the whole matter, and a triumphant reply to the objections of M. Freret. This answer called into the field a fresh antagonist, Father Soueiet, whose five dissertations on this subject were chiefly remarkable for the want of knowledge and want of decorum, which they displayed. In consequence of these discussions, Newton was in duced to prepare his larger work for the press, and had nearly completed it at the time of his death. It was published in 1728, under the title of THE CHRONOLOGY OF THE ANCIENT KINGDOMS AMENDED, TO WHICH is PREFIXED A SHORT CHRONICLE FROM THE FIRST MEMORY OF THINGS IN EUROPE TO THE CONQUEST OF PERSIA BY ALEXANDER THE GREAT. It consists of six chap ters: 1. On the Chronology of the Greeks; according to Whis- ton, our author wrote out eighteen copies of this chapter with his own hand, differing little from one another. 2. Of the Empire of Egypt; 3. Of the Assyrian Empire; 4. Of the two contempo rary Empires of the Babylonians and Medes ; 5. A Description of the Temple of Solomon ; 6. Of the Empire of the Persians ; this chapter was not found copied with the other five, but as it was discovered among his papers, arid appeared to be a continu ation of the same work, the Editor thought proper to add it thereto. Newton's LETTER TO A PERSON OF DISTINCTION WHO HAD DESIRED HIS OPINION OF THE LEARNED BlSHO^ LLOYD'S HYPOTHESIS CONCERNING THE FORM OF THE MOST ANCIENT ^EAR, closes this enumeration of his Chronological Writings. A ihird edition of the PRINCIPIA appeared in 1726, with many changes and additions. About four years were consumed in its preparation and publication, which were under the superintend- ance of Dr. Henry Pemberton, an accomplished mathematician, and the author of "A VIEW OF SIR ISAAC NEWTON'S PHILO SOPHY." 1728. This gentleman enjoyed numerous opportunities of conversing with the aged and illustrious author. " I found," says Pemberton, " he had read fewer of the modern mathemati cians than one could have expected; but his own prodigious invention readily supplied him with what he might have an occa sion for in the pursuit of any subject he undertook. I have often LIFE OF SIR ISAAC NEWTON. 57 heard him censure the handling geometrical subjects ly algebraic calculations ; and his book of Algebra he called by the name of Universal Arithmetic, in opposition to the injudicious title of Geometry, which Descartes had given to the treatise, wherein he shows how the geometer may assist his invention by such kind of computations. He thought Huygens the most elegant of any mathematical writer of modern times, and the most just imitator of the ancients. Of their taste and form of demonstration, Sir Isaac always professed himself a great admirer. I have heard him even censure himself for not following them yet more closely than he did ; and speak with regret of his mistake at the begin ning of his mathematical studies, in applying himself to the works of Descartes and other algebraic writers, before he had considered the elements of Euclid with that attention which so excellent a writer deserves." " Though his memory was much decayed," continues Dr. Pem- berton, "he perfectly understood his own writings." And even this failure of memory, we would suggest, might have been more apparent than real, or, in medical terms, more the result of func tional weakness than organic decay. Newton seems never to have confided largely to his memory : and as this faculty mani fests the most susceptibility to cultivation ; so, in the neglect of due exercise, it more readily and plainly shows a diminution of its powers. Equanimity and temperance had, indeed, preserved Newton singularly free from all mental and bodily ailment. His hair was, to the last, quite thick, though as white as silver. He never made use of spectacles, and lost but one tooth to the day of his death. He was of middle stature, well-knit, and, in the latter part of his life, somewhat inclined to be corpulent. Mr. Conduit says, " he had a very lively and piercing eye, a comely and gra cious aspect, with a fine head of hair, white as silver, without any baldness, and when his peruke was off was a venerable sight." According to Bishop Atterbury, "in the whole air of his face and make there was nothing of that penetrating sagacity which appears in his compositions. He had something rather languid in his look and manner which did not raise any great expectation 58 LIFE OF SIR ISAAC NEWTON. in those who did not know him." Hearne remarks, " Sir Isaac was a man of no very promising aspect. He was a short, well- set man. He was full of thought, and spoke very little in com pany, so that his conversation was not agreeable. When he rode in his coach, one arm would be out of his coach on one side and the other on the other." These different accounts we deem easily reconcilable. In the rooms of the Royal Society, in the street, or in mixed assemblages, Newton's demeanour — always courteous, unassuming and kindly — still had in it the overawings of a profound repose and reticency, out of which the communica tive spirit, and the "lively and piercing eye" would only gleam in the quiet and unrestrained freedom of his own fire-side. " But this I immediately discovered in him," adds Pemberton, still further, "which at once both surprised and charmed me. Neither his extreme great age, nor his universal reputation had rendered him stiff in opinion, or in any degree elated. Of this I had occasion to have almost daily experience. The remarks I continually sent him by letters on his Principia, were received with the utmost goodness. These were so far from being any ways displeasing to him, that, on the contrary, it occasioned him to speak many kind things of me to my friends, and to honour me with a public testimony of his good opinion." A modesty, open ness, and generosity, peculiar to the noble and comprehensive spirit of Newton. " Full of wisdom and perfect in beauty," yet not lifted up by pride nor corrupted by ambition. None, how ever, knew so well as himself the stupendousness of his discoveries in comparison with all that had been previously achieved ; and none realized so thoroughly as himself the littleness thereof in comparison with the vast region still unexplored. A short time before his death he uttered this memorable sentiment: — " I do not know what I may appear to the world ; but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, while the great ocean of truth lay all undis covered before me." How few ever reach the shore even, much less find "a smoother pebble or a prettier shell!" Newton had now resided about two years at Kensington ; and LIFE OF SIR ISAAC NEWTON. 59 the air which he enjoyed there, and the state of absolute rest, proved of great benefit to him. Nevertheless he would occasion ally go to town. And on Tuesday, the 28th of February, 1727, he proceeded to London, for the purpose of presiding at a meeting of the Royal Society. At this time his health was considered, by Mr. Conduit, better than it had been for many years. But the unusual fatigue he was obliged to suffer, in attending the meeting, and in paying and receiving visits, speedily produced a violent return of the affection in the bladder. He returned to Kensington on Saturday, the 4th of March. Dr. Mead and Dr. Cheselden attended him ; they pronounced his disease to be the stone, and held out no hopes of recovery. On Wednesday, the 15th of March, he seemed a little better; and slight, though groundless, encouragement was felt that he might survive the attack. From the very first of it, his sufferings had been intense. Paroxysm followed paroxysm, in quick succession : large drops )f sweat rolled down his face ; but not a groan, not a complaint, not the least mark of peevishness or impatience escaped him : and during the short intervals of relief, he even smiled and con versed with his usual composure and cheerfulness. The flesh quivered, but the heart quaked not ; the impenetrable gloom was settling down : the Destroyer near ; the portals of the tomb opening, still, arnid this utter wreck and dissolution of the mortal, the immortal remained serene, unconquerable : the radiant light broke through the gathering darkness ; and Death yielded up its sting, and the grave its victory. On Saturday morning, 18th, he read the newspapers, and carried on a pretty long conversation with Dr. Mead. His senses and faculties were then strong and vigorous ; but at six o'clock, the same evening, he became insen sible ; and in this state he continued during the whole of Sunday, and till Monday, the 20th, when he expired, between one and two o'clock in the morning, in the eighty-fifth year of his age. And these were the last days of Isaac Newton. Thus closed the career of one of earth's greatest and best men. His mission was fulfilled. Unto the Giver, in many-fold addition, the talents were returned. While it was yet day he had worked ; and for the night that quickly cometh he was not unprepared. Full of 60 LIFE OF SIR ISAAC NEWTON. years, ind full of honours, the heaven-sent was recalled ; and, in the confidence of a " certain hope," peacefully he passed awa}' into the silent depths of Eternity. His body was placed in Westminster Abbey, with the state and ceremonial that usually attended the interment of the most distinguished. In 1731, his relatives, the inheritors of his personal estate, erected a monument to his memory in the most conspicu ous part of the Abbey, which had often been refused by the dean and chapter to the greatest of England's nobility. During the same year a medal was struck at the Tower in his honour ; arid, in 1755, a full-length statue of him, in white marble, admirably executed, by Roubiliac, at the expense of Dr. Robert Smith, was erected in the ante-chamber of Trinity College, Cambridge. There is a painting executed in the glass of one of the windows of the same college, made pursuant to the will of Dr. Smith, who left five hundred pounds for that purpose. Newton left a personal estate of about thirty-two thousand pounds. It was divided among his four nephews and four nieces of the half blood, the grand-children of his mother, by the Reve rend Mr. Smith. The family estates of Woolsthorpe arid Sustern fell to John Newton, the heir-at-law, whose great grand-father was Sir Isaac's uncle. Before his death he made an equitable distribution of his two other estates : the one in Berkshire to the sons and daughter of a brother of Mrs. Conduit ; and the other, at Kensington, to Catharine, the only daughter of Mr. Conduit, and who afterward became Viscountess Lymington. Mr. Con duit succeeded to the offices of the Mint, the duties of which he had discharged during the last two years of Sir Isaac's life. Our author's works are found in the collection of Castilion, Berlin, 1744, 4to. 8 torn.; in Bishop Horsley's Edition, London, 1779, 4to. 5 vol.; in the Biographia Brittannica, &c. Newton also published Bern. Varcnii Geographia, &c., 1681, 8vo. There are, however, numerous manuscripts, letters, and other papers, which have never been given to the world: these are preserved, in various collections, namely, in the library of Trinity College, Cambridge ; in the library of Corpus Christi College, Oxford ; in the library of Lord Macclesfield : and, lastly arid LIFE OF SIR ISAAC NEWTON. 61 chiefly, in the possession of the family of the Earl of Portsmouth, through the Viscountess Lymington. Everything appertaining to Newton has been kept and che rished with peculiar veneration. Different memorials of him are preserved in Trinity College, Cambridge ; in the rooms of the Royal Society, of London : and in the Museum of the Royal Society of Edinburgh. The manor-house, at Woolsthorpe, was visited by Dr. Stuke ley, in October, 1721, who, in a letter to Dr. Mead, written in 1727, gave the following description of it: — " 'Tis built of stone, as is the way of the country hereabouts, and a reasonably good one. They led me up stairs and showed me Sir Isaac's stud}-, where I supposed he studied, when in the country, in his younger days, or perhaps when he visited his mother from the University. I observed the shelves were of his own making, being pieces of deal boxes, which probably he sent his books and clothes down in on those occasions. There were, some years ago, two or threr hundred books in it of his father-in-law, Mr. Smith, which Sir Isaac gave to Dr. Newton, of our town." The celebrated apple- tree, the fall of one of the apples of which is said to have turned the attention of Newton to the subject of gravity, was destroyed by the wind about twenty years ago ; but it has been preserved in the form of a chair. The house itself has been protected with religious care. It was repaired in 1798, and a tablet of white marble put up in the room where our author was born, with the follow, ng inscription : — " Sir Isaac Newton, son of John Newton, Lord of the Manor of Woolsthorpe, was born in this room, on the 25th of December, 1642." Nature and Nature's Laws wei-e hid in night, God said, " Let NEWTON be," and all was light. THE PEINCIPIA. THE AUTHOR'S PREFACE SINCE the ancients (as we are told by Pappus), made great account oi the science of mechanics in the investigation of natural things : and the moderns, laying aside substantial forms and occult qualities, have endeav oured to subject the phenomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy. The ancients considered mechanics in a twofold respect ; as rational, which proceeds accurately by demonstration ; and practical. To practical me chanics all the manual arts belong, from which mechanics took its name. Rut as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accu rate is called geometrical , what is less so, is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic ; and if any could work with perfect accuracy, he would be the most perfect mechanic of all ; for the description if right lines and circles, upon which geometry is founded, belongs to me chanics. Geometry does not teach us to draw these lines, but requires them to be drawn ; for it requires that the learner should f.rst be taught to describe these accurately, before he enters upon geometry ; then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics ; and by geometry the use of them, when so solved, is shown ; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things. Therefore geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that geometry is commonly referred to their magnitudes, and mechanics to their motion. In this sense rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any mo tions, accurately proposed and demonstrated. This part of mechanics was i:;vm THE AUTHOR & PREFACE. cultivated by the ancients in the five powers which relate to manual arts, who considered gravity (it not being a manual power), ho Otherwise than as it moved weights by those powers. Our design not respecting arts, but philosophy, and our subject not manual but natural powers, we consider chiefly those things which relate to gravity, levity, elastic force, the resist ance of fluids, and the like forces, whether attractive or impulsive ; and therefore we offer this work as the mathematical principles :f philosophy ; for all the difficulty of philosophy seems to consist in this — from the phenom ena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena ; and to this end the general propositions in the first and second book are directed. In the third book we give an example of this in the explication of the System of the World : for by the propositions mathematically demonstrated in the former books, we in the third derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, we deduce the mo tions of the planets, the comets, the moon, and the sea. I wish we could do- rive the rest of the phenomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies. by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto at tempted the search of nature in vain ; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy. In the publication of this work the most acute and universally learned Mr. Edmund H alley not only assisted me with his pains in correcting the press and taking care of the schemes, but it was to his solicitations that its becoming public is owing ; for when he had obtained of me my demonstra tions of the figure of the celestial orbits, he continually pressed me to com municate the same to the Royal Societ //, who afterwards, by their kind en couragement and entreaties, engaged me to think of publishing them. But after I had begun to consider the inequalities of the lunar motions, and had entered upon some other things relating to the laws and measures oi gravity, and other forces : and the figures that would be described by bodies attracted according to given laws ; and the motion of several bodies moving among themselves; the motion of bodies in resisting mediums; the forces, densities, and motions, of rn( Hums ; the orbits of the comets, and such like ; Ixix deferred that publication till I had made a searcli into those matters, and could put forth the whole together. What relates to the lunar motions (be ing imperfect), I have put all together in the corollaries of Prop. 66, to avoid being obliged to propose and distinctly demonstrate the several things there contained in a method more prolix than the subject deserved, and in terrupt the series of the several propositions. Some things, found out after the rest, I chose to insert in places less suitable, rather than change the number of the propositions and the citations. I heartily beg that what 1 have here done may be read with candour; and that the defects in a subject so difficult be not so much reprehended as kindly supplied, and in vestigated by new endeavours of mv readers. ISAAC NEWTON. Cambridge, Trinity Coupge May 8, l«iHB. In the second edition the second section of the first book was enlarged. In the seventh section of the second book the theory of the resistances of fluids was more accurately investigated, and confirmed by new experiments. In the third book the moon's theory and the profession of the equinoxes were more fully deduced from their principles ; and the theory of the comets was confirmed by more examples of the calculati >n of their orbits, done also with greater accuracy. In this third edition the resistance of mediums is somewhat more largely handled than before; and new experiments of the resistance of heavy bodies falling in air are added. In the third book, the argument to prove that the moon is retained in its orbit by the force of gravity is enlarged on ; and there are added new observations of Mr. Pound's of the proportion of the diameters of Ju.piter to each other : there are, besides, added Mr. Kirk's observations of the comet in 16SO ; the orbit of that comet com puted in an ellipsis by Dr. Halley ; and the ortit of the comet in computed by Mr. Bradley, OOK I. THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY DEFINITIONS. DEFINITION I. 77w? quantity of matter is the measure of the same, arising from its density and hulk conjutictly. THUS air of a double density, in a double space, is quadruple in quan- ti ty ; in a triple space, sextuple in quantity. The same thing is to be un derstood of snow, and fine dust or powders, that are condensed by compres sion or liquefaction • and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts oi bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body ; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter. DEFINITION II. The quantity of motion is the measure nf tlie same, arising from the velocity and quantity of matter corjunctly. The motion of the whole i<! the sum of the motions of all the parts ; and therefore in a body double in quantity, with equal velocity, the motion is iouble ; with twice the velocity, it is quadruple, DEFINITION III. The vis insita, or innate force of matter, is a power of resisting, hy which every body, as much as in it lies, endeavours to persevere in its present stale, whether it be of rest, or of moving uniformly forward in a right line. This force is ever proportional to the body whose force it is ; and differs nothing from the inactivity of the mass, but in our manner of conceiving T4 THE MATHEMATICAL PRINCIPLES it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertia, or force of inactivity. Hut a body exerts this force only, when another force, impressed upon it, endeavours to change its condition ; and the exercise of this force may bo considered both as resistance and impulse ; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distin guished ; nor are those bodies always truly at rest, which commonly are taken to be so. DKFLMTIOX IV. Ait impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line. This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis inertice only. Impressed forces are of differe.it origins • as from percussion, from pressure, from centripetal force. DEFINITION V. A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre. Of this sort is gravity, by which bodies tend to the centre of the earth magnetism, by which iron tends to the loadstone ; and that force, what ever it is, by which the planets are perpetually drawn aside from the rec tilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone, whirled about in a sling, endeavours to re cede from the hand that turns it ; and by that endeavour, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and as soon as ever it is let go, flies away. That force which opposes itself to this endeavour, and by which the sling perpetually draws back the stone towards the hand, and retains it in its orbit, because it is directed to the hand as the centre of the orbit, I call the centripetal force. And the same thing is to be understood of all bodies, revolved in any orbits. They all endeavour to recede from the centres of their orbits ; and wore it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would tiy off in right lines, with an uniform motion. A projectile, if it was not for the force of gravity, would not deviate towards the earth, tut would OF NATUJIAL PHILOSOPHY. 7fl go off from it in a right line, and that with an uniform motion,, if the re sistance of the air was taken away. It is by its gravity that it is drawn aside perpetually from its rectilinear course, and made to deviate towards the earth, more or less, according to the force of its gravity, and the velo city of its motion. The less its gravity is, for the quantity of its matter, or the greater the velocity with which it is projected, the less will it devi ate from a rectilinear course, and the farther it will go. If a leaden balJ, projected from the top of a mountain by the force of gunpowder with a given velocity, and in a direction parallel to the horizon, is carried in a curve line to the distance of two miles before it falls to the ground ; the same, if the resistance of the air were taken away, with a double or decuple velocity, would fly twice or ten times as far. And by increasing the velo city, we may at pleasure increase the distance to which it might be pro jected, and diminish the curvature of the line, which it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls ; or lastly, so that it might never fall to the earth, but go forward into the celestial spaces, and pro ceed in its motion in iiifiuitum. And after the same manner that a pro jectile, by the force of gravity, may be made to revolve in an orbit, and go round the whole earth, the moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impels it towards the earth, may be perpetually drawn aside towards the earth, out of the r&ti- linear way, which by its innate force it would pursue; and would be made to revolve in the orbit which it now describes ; nor could the moon with out some such force, be retained in its orbit. If this force was too small, it would not sufficiently turn the moon out of a rectilinear course : if it was too great, it would turn it too much, arid draw down the moon from its orbit towards the earth. It is necessary, that the force be of a just quantity, and it belongs to the mathematicians to find the force, that may serve exactly to retain a body in a given orbit, with a given velocity ; and vice versa, to determine the curvilinear way, into which a body projected from a given place, with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force. The quantity of any centripetal force may be considered as of three kinds; aboolu'e, accelerative, and motive. DEFINITION VI. The absolute quantity of a centripetal force is the measure f>f the same proportional to the efficacy of the cause that propagates it from the cen tre, through the spaces round about. Thus the magnetic force is greater in one load-stone and less in another according to their sizes and strength of intensity. 76 THE MATHEMATICAL PRINCIPLES DEFINITION VII. The accelerative quantity of a centripetal force is the measure, of tht same, proportional to the velocity which it generates in a given time. Thus the force of the same load-stone is greater at a less distance, and less at a greater : also the force of gravity is greater in valleys, less on tops of exceeding high mountains ; and yet less (as shall hereafter be shown), at greater distances from the body of the earth ; but at equal distan ces, it is the same everywhere ; because (taking away, or allowing for, the resistance of the air), it equally accelerates all falling bodies, whether heavy or light, great or small. DEFINITION VIII. TJie motive quantity of a centripetal force, is the measure of the samt\ proportional to the motion which it generates in a given twip. Thus the weight is greater in a greater body, less in a less body ; and. in the same body, it is greater near to the earth, and less at remoter dis tances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or, as I may say, its weight ; and it is al ways known by the quantity of an equal and contrary force just sufficient to Ifinder the descent of the body. These quantities of forces, we may, for brevity's sake, call by the names of motive, accelerative, and absolute forces ; and, for distinction's sake, con sider them, with respect to the bodies that tend to the centre ; to the places of those bodies ; and to the centre of force towards which they tend ; that is to say, I refer the motive force to the body as an endeavour and propen sity of the whole towards a centre, arising from the propensities of the several parts taken together ; the accelerative force to the place of the body, as a certain power or energy diffused from the centre to all places around to move the bodies that are in them : and the absolute force to the centre, as endued with some cause, without which those motive forces would not be propagated through the spaces round about ; whether that cause be some central body (siuh as is the load-stone, in the centre of the magnetic force, or the earth in the centre of the gravitating force), or anything else that does not yet appear. For I here design only to give a mathematical notion of those forces, without considering their physical causes and seats. Wherefore the accelerative force will stand in the same relation to the motive, as celerity does to motion. For the quantity of motion arises from the celerity drawn into the quantity of matter : and the motive force arises from the accelerative force drawn into the same quantity of matter. For the sum of the actions of the accelerative force, upon the several ; articles of the body, is the motive force of the whole. Hence it is, that near the OF NATURAL PHILOSOPHY. 77 surface of the earth, where the accelerative gravity, or force productive of gravity, in all bodies is the same, the motive gravity or the weight is as the body : but if we should ascend to higher regions, where the accelerative gravity is less, the weight would be equally diminished, and would always be as the product of the body, by the accelerative gravity. So in those re gions, where the accelerative gravity is diminished into one half, the weight of a body two or three times less, will be four or six times less. I likewise call attractions and impulses, in the same sense, accelerative, and motive ; and use the words attraction, impulse or propensity of any sort towards a centre, promiscuously, and indifferently, one for another ; considering those forces not physically, but mathematically : wherefore, the reader is not to imagine, that by those words, I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points) ; when at any time I happen to speak of centres as attracting, or as endued with attractive powers. SCHOLIUM. Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be under stood in the following discourse. I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the re moving of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common. I. Absolute, true, and mathematical time, of itself, and from its own na ture flows equably without regard to anything external, and by another name is called duration : relative, apparent, and common time, is some sen sible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ; such as an hour, a day, a month, a year. II. Absolute space, in its own nature, without regard to anything exter nal, remains always similar and immovable. Relative space is some mo vable dimension or measure of the absolute spaces ; which our senses de termine by its position to bodies ; and which is vulgarly taken for immo vable space ; such is the dimension of a subterraneous, an aereal, or celestial space, determined by its position in respect of the earth. Absolute and relative space, are the same in figure and magnitude ; but they do not re main always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains al ways the same, will at one time be one part of the absolute space into which TS THE MATHEMATICAL PRINCIPLES the air passes ; at another time it will be another part of the same, and so. absolutely understood, it will be perpetually mutable. III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space ; not the situation, nor the external surface of the body. For the places of equal solids are always equal ; but their superfices, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same thing with the sum of the motions of the parts ; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places ; and therefore the place of the whole is the same thing with the sum of the places of the parts, and for that reason, it is internal, and in the whole body. IV. Absolute motion is the translation of a body from one absolute place into another ; and relative motion, the translation from one relative place into another. Thus in a ship under sail, the relative place of a body is that part of the ship which the body possesses ; or that part of its cavity which the body fills, and which therefore moves together with the ship : and relative rest is the continuance of the body in the same part of the ship, or of its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space, in which the ship itself, its cavity, and all that it contains, is moved. Wherefore, if the earth is really at rest, the body, which relatively rests in the ship, will really and absolutely move with the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body will arise, partly from the true motion of the earth, in immovable space ; partly from the relative motion of the ship on the earth ; and if the body moves also relatively in the ship ; its true motion will arise, partly from the true motion of the earth, in immovable space, and partly from the relative mo tions as well of the ship on the earth, as of the body in the ship ; and from these relative motions will arise the relative motion of the body on the earth. As if that part of the earth, where the ship is, was truly moved toward the east, with a velocity of 10010 parts; while the ship itself, with a fresh gale, and full sails, is carried towards the west, with a velocity ex pressed by 10 of those parts ; but a sailor walks in the ship towards the east, with 1 part of the said velocity ; then the sailor will be moved truly in immovable space towards the east, with a velocity of 10001 parts, and relatively on the earth towards the west, with a velocity of 9 of those parts. Absolute time, in astronomy, is distinguished from relative, by the equa tion or correction of the vulgar time. For the natural days are tr^y un equal, though they are commonly considered as equal, and used for a meas ure of time ; astronomers correct this inequality for their more accurate deducing of the celestial motions. It may be, that there is no such thing as an equable motion, whereby time may H accurately measured. All mo OF NATURAL PHILOSOPHY. 79 tions may be accelerated and retarded; but the true, or equable, progress of absolute time is liable to no change. The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all : and therefore it ought to be distinguished from what are only sensible measures thereof ; and out of which we collect it, by means of the astronomical equation. The necessity of which equation, for deter mining the times of a phamomenon, is evinced as well from the experiments of the pendulum clock, as by eclipses of the satellites of Jupiter. As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they will be moved (if the expression may be allowed) out of themselves. For times and spaces are, as it were, the places as well of themselves as of all other things. All things are placed in time as to order of succession ; and in space as _to order of situation. It is from their essence or nature that they are places ; and that the primary places of things should be moveable, is absurd. These are therefore the absolute places ; and trans lations out of those places, are the only absolute motions. But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body con sidered as immovable, we define all places ; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And'so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common af fairs ; but in philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred. But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest ; but impossi ble to know, from the position of bodies to one another in our regions whether any of these do keep the same position to that remote body; it follows that absolute rest cannot be determined from the position of bodies in our regions. It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavour to recede from the axis of motion ; and the impetus of bodies moving forward, arises from the joint impetus of all the parts. Therefore, if surrounding bodies are moved, those that are relatively at rest within them, will partake of their motion. Upon which account, the true and absolute motion of a body cannot be Jeter- 8C THE MATHEMATICAL PRINCIPLES mined by the translation of it from those which only seem to rest ; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise, all included bodies, beside their translation from near the surrounding ones, partake likewise of their true motions ; and though that translation were not made they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded as the exterior part of a whole does to the interior, or as the shell does to the kernel ; but, if the shell moves, the kernel will also move, as being part of the whole, without any removal from near the shell. A property, near akin to the preceding, is this, that if a place is moved, whatever is placed therein moves along with it ; and therefore a body, which is moved from a place in motion, partakes also of the motion of its place. Upon which account, all motions, from places in motion, are no other than parts of entire and absolute motions ; and every entire motion is composed of the motion of the body out of its first place, and the motion of this place out of its place ; and so on, until we come to some immovable place, as in the before-mentioned example of the sailor. Where fore, entire and absolute motions can be no otherwise determined than by immovable places : and for that reason I did before refer those absolute motions to immovable places, but relative ones to movable places. Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another ; and upon this account must ever remain unmoved ; and do thereby constitute immovable space. The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved : but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress some force on other bodies with which the former is compared, that by their giving way, that relation may be changed, in which the re lative rest or motion of this other body did consist. Again, true motion suffers always some change from any force impressed upon the moving body ; but relative motion docs not necessarily undergo any change by such forces. For if the same forces are likewise impressed on those other bodies, with which the comparison is made, that the relative position may be pre served, then that condition will be preserved in which the relative motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be preserved when the true suffers some change. Upon which accounts; true motion does by no means consist in such relations. The effects whicli distinguish absolute from relative motion arc, the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute cir cular motion., they are greater or less, according t » the quantity of the OF NATURAL PHILOSOPHY. «1 motion. If a vessel, hung: by & }ong cord, is so often turned ubout that the cord is strongly twisted, then filled with water, and held at rest together with the water ; after, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for some time in this motion ; the surface of the water will at first be plain, as before the vessel began to move : but the vessel; by grad ually communicating its motion to the water, will make it begin sensibly ^to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure (as I have experi enced), and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its en deavour to recede from the axis of its motion ; and the true and absolute circular motion of the water, which is here directly contrary to the rela- tivej discovers itself, and may be measured by this endeavour. At first, when the relative motion of the water in the vessel was greatest, it pro duced no endeavour to recede from the axis ; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its en deavour to recede from the axis ; and this endeavour showed the real cir cular motion of the water perpetually increasing, till it had acquired its greatest quantity, wh en the water rested relatively in the vessel. And therefore this endeavour does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion, as its proper and adequate effect ; but relative motions, in one and the same body, are innumerable, according to the various relations it bears to external bodies, and like other relations, arc altogether destitute of any real effect, any otherwise than they may perhaps par take of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them ; the several parts of those heavens, and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest), and being carried together with their heavens, partake of their motions, and as parts of revolving wholes, endeavour to recede from the axis of their motions. Wherefore relative quantities are not the quantities themselves, whose names they bear, but those sensible measures of them (either accurate cr inaccurate), which arc commonly used instead of the measured quantities themselves. And if the meaning of words is to he determined bv their 82 THE MATHEMATICAL PRINCIPLES use, then by the names time, space, place and motion, their measures arv properly to be understood ; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. Upon which account, they do strain the sacred writings, who there interpret those words for the measured quantities. Nor do those less defile the purity of mathematical and philosophical truths, who confound real quan tities themselves with their relations and vulgar measures. It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent ; be cause the parts of that immovable space, in which those motions are per formed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate : for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions ; partly from the forces, which are the causes and effects of the true motions. For instance, if tAvo globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavour of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decr» ase of the tensicn of 1 le cord, we might infer the increment or decrement of their motions : and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented ; that is, we might discover their hinder- most faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and the determination of this circu lar motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared. But now, if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions, we could not indeed determine from the relative translation of the globes among those bodies, whether the motion did belong to the globes or to the bodies. But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest ; and then, lastly, from the trans lation of the globes among the bodies, we should find the determination oi their motions. But how we are to collect the true motions from their causes, effects, and apparent differences ; and, vice versa, how from the mo tions, either true or apparent, we may come to the knowledge of theii causes and effects, shall be explained more at large in the following tra<;t For to this end it was that I composed it. OF NATURAL PHILOSOPHY. AXIOMS, OR LAWS OF MOTION. LAW I. Hvery body perseveres in its state of rest, or of uniform motion in a ri^ht line, unless it is compelled to change that state by forces impressed thereon. PROJECTILES persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is re tarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve then jDotions both pro gressive and circular for a much longer time. LAW II. The alteration of motion is ever proportional to the motive force imp reus ed ; and is made in the direction of the right line in. which that force is impressed. If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other ; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both. LAW III. To every action there is always opposed an equal reaction : or the mu tual actions of two bodies upon each other are always equal, and di rected to contrary parts. Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. 84 THE MATHEMATICAL PRINCIPLES If a body impinge upon another, and by its force change the motion of (It* other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies ; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally pro portional to the bodies. This law takes place also in attractions, as will be proved in the next scholium. COROLLARY I. A body by two forces conjoined will describe the diagonal of a parallelo gram, in the same time that it wovld describe the sides, by those forces apart. If a body in a given time, by the force M impressed apart in the place A, should with an uniform motion / be carried from A to B ; and by the force N impressed apart in the same place, should be carried from A to c ~\) C ; complete the parallelogram ABCD, and, by both forces acting together, it will in the same time be carried in the diagonal from A to D. For since the force N acts in the direction of the line AC, parallel to BD, this force (by the second law) will not at all alter the velocity generated by the other force M, by which the body is carried towards the line BD. The body therefore will arrive at the line BD in the same time, whether the rorce N be impressed or not ; and therefore at the end of that time it will he found somewhere in the line BD. By the same argument, at the end of the same time it AY ill be found somewhere in the line CD. Therefore it will be found in the point D, where both lines meet. But it will move in ;i right line from A to D, by Law I. COROLLARY II. And hence is explained the composition of any one direct force AD, out of any two oblique forces AC and CD ; and, on the contrary, the re solution of any one direct force AD into two oblique forces AC and CD : which composition and resolution are abundantly confirmed from, mechanics. As if the unequal radii OM and ON drawn from the centre O of any wheel, should sustain the weights A and P by the cords MA and NP ; and the forces of those weights to move the wheel were required. Through the rentre O draw the right line KOL, meeting the cords perpendicularly in A and L; and from the centre O, with OL the greater of the distances OF NATURAL PHILOSOPHY. OK arid OL, describe a circle, meeting the cord MA in D : and drawing OD, make AC paral- "^ lei and DC perpendicular thereto. Now, it being indifferent whether the points K, L, D, of the cords be lixed to the plane of the wheel or not, the weights will have the same effect whether they are suspended from the points K and L, or from D and L. Let the whole force of the weight A be represented by the line AD, and let it be resolved into the forces AC and CD ; of which the force AC, drawing the radius OD directly from the centre, will have no effect to move the wheel : but the other force DC, drawing the radius DO perpendicularly, will have the same effect as if it drew perpendicularly the radius OL equal to OD ; that is, it w ill have the same effect as the weight P, if that weight is to the weight A as the force DC is to the force DA ; that is (because of the sim ilar triangles ADC, DOK), as OK to OD or OL. Therefore the weights A and P, which are reciprocally as the radii OK and OL that lie in the same right line, will be equipollent, and so remain in equilibrio ; which is the well known property of the balance, the lever, and the wheel. If either weight is greater than in this ratio, its force to move the wheel will be so much greater. If the weight p, equal to the weight P, is partly suspended by the cord NJO, partly sustained by the oblique plane pG ; draw p}i, NH, the former perpendicular to the horizon, the latter to the plane pG ; and if the force of the weight p tending downwards is represented by the line /?H, it may be resolved into the forces joN, HN. If there was any plane /?Q, perpendicular to the cord y?N, cutting the other plane pG in a line parallel to the horizon, and the weight p was supported only by those planes pQ, pG, it would press those planes perpendicularly with the forces pN, HN; to wit, the plane joQ, with the force joN, and the plane pG with the force HN. And therefore if the plane pQ was taken away, so thnt the weight might stretch the cord, because the cord, now sustaining the weight, supplies the place of the plane that was removed, it will be strained by the same force joN which pressed upon the plane before. Therefore, the tension of this oblique cord joN will be to that of the other perpendic ular cord PN as jt?N to joH. And therefore if the weight p is to the weight A in a ratio compounded of the reciprocal ratio of the least distances of the cords PN, AM, from the centre of the wheel, and of the direct ratio of pH tojoN, the weights will have the same effect towards moving the wheel, and will therefore sustain each other : as any one may find by experiment. But the weight p pressing upon those two oblique planes, may be con sidered as a wedge between the two internal surfaces of a body split by it; and hence tlif ft IV.P* of th^ v, ^dge and the mallet may be determined; foi 8G THE MATHEMATICAL PRINCIPLES because the force with which the weight p presses the plane pQi is to the force with which the same, whether by its own gravity, or by the blow of a mallet, is impelled in the direction of the line joH towards both the planes, as joN to pH ; and to the force with which it presses the other plane pG, as joN to NH. And thus the force of the screw may be deduced from a like resolution of forces ; it being no other than a wedge impelled with the force of a lever. Therefore the use of this Corollary spreads far and wide, and by that diffusive extent the truth thereof is farther con firmed. For on what has been said depends the whole doctrine of mechan ics variously demonstrated by different authors. For from hence are easily deduced the forces of machines, which are compounded of wheels, pullics, levers, cords, and weights, ascending directly or obliquely, and other mechan ical powers ; as also the force of the tendons to move the bones of animals. COROLLARY III. The (/uaittity of motion, which is collected by taking the sum of the mo tions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action oj bodies among themselves. For action and its opposite re-action are equal, by Law III, and there fore, by Law II, they produce in the motions equal changes towards oppo site parts. Therefore if the motions are directed towards the same parts. whatever is added to the motion of the preceding body will be subducted from the motion of that which follows ; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both ; and therefore the difference of the motions directed towards opposite parts will remain the same. Thus if a spherical body A with two parts of velocity is triple of a spherical body B which follows in the same right line with ten parts of velocity, the motion of A will be to that of B as 6 to 10. Suppose, then, their motions to be of 6 parts and of 10 parts, and the sum will be 16 parts. Therefore, upon the meeting of the bodies, if A acquire 3, 4, or 5 parts of motion, B will lose as many ; and therefore after reflexion A will proceed With 9, 10, or 11 parts, and B with 7, 6, or 5 parts; the sum remaining always of 16 parts as before. If the body A acquire 9, 10, 11, or 12 parts of motion, and therefore after meeting proceed with 15, 16, 17, or 18 parts, the body B, losing so many parts as A has got, will either proceed with 1 part, having lost 9, or stop and remain at rest, as having lost its whole progressive motion of 10 parts ; or it will go back with 1 part, having not only lost its whole motion, but (if 1 may so say) one part more; or it will go back with 2 parts, because a progressive mo tion of 12 parts is taken off. And so the sums of the Conspiring motions 15 ,1, or 16-1-0, and the differences of the contrary i otions 17 — 1 and OF NATURAL PHILOSOPHY. [S — 2, will always be equal to 16 parts, as they were before tie meeting and reflexion of the bodies. But, the motions being known with whicli the bodies proceed after reflexion, the velocity of either will be also known, by taking the velocity after to the velocity before reflexion, as the motion after is to the motion before. As in the last case, where the motion of tho body A was of 0 parts before reflexion and of IS parts after, and the velocity was of 2 parts before reflexion, the velocity thereof after reflexion will be found to be of 6 parts ; by saying, as the 0 parts of motion before to 18 parts after, so are 2 parts of velocity before reflexion to (5 parts after. But if the bodies are cither not spherical, or, moving in different right lines, impinge obliquely one upon the other, and their mot'ons after re flexion are required, in those cases we are first to determine the position of the plane that touches the concurring bodies in the point of concourse , then the motion of each body (by Corol. II) is to be resolved into two, one perpendicular to that plane, and the other parallel to it. This done, be cause the bodies act upon each other in the direction of a line perpendicu lar to this plane, the parallel motions are to be retained the same after reflexion as before ; and to the perpendicular motions we are to assign equal changes towards the contrary parts ; in such manner that the sum of the conspiring and the difference of the contrary motions may remain the same as before. From such kind of reflexions also sometimes arise the circular motions of bodies about their own centres. But these are cases which I do not consider in what follows ; and it would be too tedious to demonstrate every particular that relates to this subject. COROLLARY IV. The common centre of gravity of two or more bodies does not alter its state of 'motion or rest by the actions of the bodies among themselves ; and therefore the common centre of gravity of all bodies acting upon each other (excluding outward actions and impediments) is either at rest, or moves uniformly in a right line. For if two points proceed with an uniform motion in right lines, and their distance be divided in a given ratio, the dividing point will be either at rest, or proceed uniformly in a right line. This is demonstrated here after in Lem. XXIII and its Corol., when the points are moved in the same plane ; and by a like way of arguing, it may be demonstrated when the points are not moved in the same plane. Therefore if any number of Kdies move uniformly in right lines, the common centre of gravity of any two of them is either at rest, or proceeds uniformly in a right line ; because the line which connects the centres of those two bodies so moving is divided at that common centre in a given ratio. In like manner the common centre of those two and that of a third body will be either at rest or moving uni formly in aright line because at that centre the distance 1 etween th«? 88 THE MATHEMATICAL PRINCIPLES common centre of the two bodies, and the centre of this last, is divided in a given ratio. In like manner the common centre of these three, and of a fourth body, is either at rest, or moves uniformly in a right line ; because the distance between the common centre of the three bodies, and the centre of the fourth is there also divided in a given ratio, and so on m itifinitum. Therefore, in a system of bodies where there is neither any mutual action among themselves, nor any foreign force impressed upon them from without, and which consequently move uniformly in right lines, the common centre of gravity of them all is either at rest or moves uniformly forward in a right line. Moreover, in a system of two bodies mutually acting upon each other, since the distances between their centres and the common centre of gravity of both are reciprocally as the bodies, the relative motions of those bodies, whether of approaching to or of receding from that centre, will be equal among themselves. Therefore since the changes which happen to motions are equal and directed to contrary parts, the common centre of those bodies, by their mutual action between themselves, is neither promoted nor re tarded, nor suffers any change as to its state of motion or rest. But in a system of several bodies, because the common centre of gravity of any two acting mutually upon each other suffers no change in its state by that ac tion : and much less the common centre of gravity of the others with which that action does not intervene ; but the distance between those two centres is divided by the common centre of gravity of all the bodies into parts re ciprocally proportional to the total sums of those bodies whose centres they are : and therefore while those two centres retain their state of motion or rest, xhe common centre of all does also retain its state : it is manifest that the common centre of all never suffers any change in the state of its mo tion or rest from the actions of any two bodies between themselves. But in such & system all the actions of the bodies among themselves either hap pen between two bodies, or are composed of actions interchanged between some two bodies ; and therefore they do never produce any alteration in the comrrv n centre of alias to its state of motion or rest. Wherefore tiince that centre, when the bodies do not act mutually one upon another, Oilier is nt rest or moves uniformly forward in some right line, it will, :v\>U7ithst?nding the mutual actions of the bodies among themselves, always jAY-jevere in its state, either of rest, or of proceeding uniformly in a right liiv,, unless it is forced out of this state by the action of some power im- prev^-d from without upon the whole system. And therefore the same law take*1 place in a system consisting of many bodies as in one single body, with wsgard to their persevering in their state of motion or of rest. For the pi \\jressive motion, whether of one single body, or of a whole system of bodies us always to be estimated from the motion of the centre of gravity. COROLLARY V. The motions cf bcdies included in a given space a ~e Ike same among OF NATURAL PHILOSOPHY. 89 themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion. For the differences of the motions tending towards the same parts, and the sums of those that tend towards contrary parts, are, at first (by sup position), in both cases the same ; and it is from those sums and differences that the collisions and impulses do arise with which the bodies mutually impinge one upon another. Wherefore (by Law II), the effects of those collisions will be equal in both cases ; and therefore the mutual motions of the bodies among themselves in the one case will remain equal to the mutual motions of the bodies among themselves in the other. A clear proof of which we have from the experiment of a ship ; where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line. COROLLARY VI. If bodies, any how moved among themselves, are urged in the direct-ton of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had been urged by no such forces. For these forces acting equally (with respect to the quantities of the DO dies to be moved), and in the direction of parallel lines, will (by Law II) move all the bodies equally (as to velocity), and therefore will never pro duce any change in the positions or motions of the bodies among themselves. SCHOLIUM. Hitherto I have laid down such principles as have been received by math ematicians, and are confirmed by abundance of experiments. By the first two Laws and the first two Corollaries, Galileo discovered that the de scent of bodies observed the duplicate ratio of the time, and that the mo tion of projectiles was in the curve of a parabola; experience agreeing with both, unless so far as these motions are a little retarded by the re sistance of the air. When a body is falling, the uniform force of its gravity acting equally, impresses, in equal particles of time, equal forces upon that body, and therefore generates equal velocities; and in the whole time impresses a whole force, and generates a whole velocity proportional to the time. And the spaces described in proportional times are as the velocities and the times conjunctly ; that is, in a duplicate ratio of the times. And when a body is thrown upwards, its uniform gravity im presses forces and takes off velocities proportional to the times ; and the times of ascending to the greatest heights are as the velocities to be taken off, and those heights are as the velocities and the times conjunetly, or ir, the duplicate ratio of the velocities. And if a body be projected in any direction, the motion arising from its projection jS compounded with the 90 THE MATHEMATICAL PRINCIPLES motion arising from its gravity. As if the body A by its motion of pio- jection alone could describe in a given time the right line AB, and with its motion of falling alone could describe in the same time the altitude AC ; complete the paralello- gram ABDC, and the body by that compounded motion will at the end of the time be found in the place D ; and the curve line AED, which that body describes, will be a parabola, to which the right line AB will be a tangent in A ; and whose ordinate BD will be as the square of the line AB. On the same Laws and Corollaries depend those things which have been demon strated concerning the times of the vibration of pendulums, and are con firmed by the daily experiments of pendulum clocks. By the same, to gether with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huv- gens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication ; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariottc soon after thought fit to explain in a treatise entirely upon that subject. But to bring this experiment to an accurate agreement with the theory, we are to have a due regard as well to the resistance of the air bodies. Let the spherical bodies CD F II as to the clastic force of the concurrin A, B be suspended by the parallel and equal strings AC, Bl), from the centres C, D. About these centres, with those intervals, describe the semicircles EAF, GBH, bisected by the radii CA, DB. Bring the body A to any point R of the arc EAF, and (withdrawing the body B) let it go from thence, and after one oscillation suppose it to return to the point V : then RV will be the retardation arising from the resistance of the air. Of this RV let ST be a fourth part, situated in the middle. to wit, so as RS and TV may be equal, and RS may be to ST as 3 to 2 then will ST represent very nearly the retardation during the descent from S to A. Restore the body B to its place: and, supjx sing the body A to be let fall from the point S, the velocity thereof in the place of re flexion A, without sensible error, will be the same as if it had descended m vacit.o from the point T. Upon which account this velocity may be represented by the chord of the arc TA. For it is a proposition well known to geometers, that the velocity of a pendulous body in the loAvest point is as the chord of the arc which it has described in its descent. Aftci OF NATUltAL PHILOSOPHY. 9 I reflexion, suppose the body A comes to the place s, and the body B to the place k. Withdraw the body B, and find the place v, from which if the body A, being let go, should after one oscillation return to the place r, st may be a fourth part of rv. so placed in the middle thereof as to leave is equal to tv, and let the chord of the arc tA represent the velocity which the body A had in the place A immediately after reflexion. For t will be the true and correct place to which the body A should have ascended, if the resistance of the air had been taken off. In the s.ime way we are to correct the place k to which the body B ascends, by finding the place I to which it should have ascended in vacuo. And thus everything may be subjected to experiment, in the same manner as if we were really placed in vacuo. These things being done, we are to take the product (if I may so say) of the body A, by the chord of the arc TA (which represents its velocity), that we may have its motion in the place A immediately before reflexion ; and then by the chord of the arc /A, that we may have its mo tion in the place A immediately after reflexion. And so we are to take the product of the body B by the chord of the arc B/, that we may have the motion of the same immediately after reflexion. And in like manner, when two bodies are let go together from different places, we are to find the motion of each, as well before as after reflexion; and then we may compare the motions between themselves, and collect the effects of the re flexion. Thus trying the thing with pendulums of ten feet, in unequal as well as equal bodies, and making the bodies to concur after a descent through large spaces, as of 8, 12, or 16 feet, I found always, without an error of 3 inches, that when the bodies concurred together directly, equal changes towards the contrary parts were produced in their motions, and, of consequence, that the action and reaction were always equal. As if the body A impinged upon the body B at rest with 9 parts of motion, and losing 7, proceeded after reflexion with 2, the body B was carried back wards with those 7 parts. If the bodies concurred with contrary motions, A with twelve parts of motion, and B with six, then if A receded with J4, B receded with 8 ; to wit, with a deduction of 14 parts of motion on each side. For from the motion of A subducting twelve parts, nothing will remain ; but subducting 2 parts more, a motion will be generated of 2 parts towards the contrary way ; and so, from the motion of the body B of 6 parts, subducting 14 parts, a motion is generated of 8 parts towards the contrary way. But if the bodies were made both to move towards the same way, A, the swifter, with 14 parts of motion, B, the slower, with 5, and after reflexion A went on with 5, B likewise went on with 14 parts ; 9 parts being transferred from A to B. And so in other cases. By the congress and collision of bodies, the quantity of motion, collected from the sum of the motions directed towards the same way, or from the difference, of those that were directed towards contrary ways, was never changed. For the error of an inch or two in measures may be easily ascribed to tht 92 THE MATHEMATICAL PRINCIPLES difficulty of executing everything with accuracy. It was not easy to let go the two pendulums so exactly together that the bodies should impinge one upon the other in the lowermost place AB ; nor to mark the places s, and ky to which the bodies ascended after congress. Nay, and some errors, too, might have happened from the unequal density of the parts of the pen dulous bodies themselves, and from the irregularity of the texture pro ceeding from other causes. But to prevent an objection that may perhaps be alledged against the rule, for the proof of which this experiment was made, as if this rule did suppose that the bodies were either absolutely hard, or at least perfectly elastic (whereas no such bodies are to be found in nature), 1 must add. that the experiments we have been describing, by no means depending upon that quality of hardness, do succeed as well in soft as in hard bodies. For if the rule is to be tried in bodies not perfectly hard, we are only to di minish the reflexion in such a certain proportion as the quantity of the elastic force requires. By the theory of Wren and Huygens, bodies abso lutely hard return one from another with the same velocity with which they meet. But this may be affirmed with more certainty of bodies per fectly elastic. In bodies imperfectly elastic the velocity of the return is to be diminished together with the elastic force ; because that force (except when the parts of bodies are bruised by their congress, or suffer some such extension as happens under the strokes of a hammer) is (as far as I can per ceive) certain and determined, and makes the bodies to return one from the other with a relative velocity, which is in a given ratio to that relative velocity with which they met. This I tried in balls of wool, made up tightly, and strongly compressed. For, first, by letting go the pendulous bodies, and measuring their reflexion, I determined the quantity of their elastic force ; and then, according to this force, estimated the reflexions that ought to happen in other cases of congress. And with this computa tion other experiments made afterwards did accordingly agree ; the balls always receding one from the other with a relative velocity, which was to the relative velocity with which they met as about 5 to 9. Balls of steel returned with almost the same velocity : those of cork with a velocity some-^ thing less ; but in balls of glass the proportion was as about 15 to 16. And thus the third Law, so far as it regards percussions and reflexions, is proved by a theory exactly agreeing with experience. In attractions, I briefly demonstrate the thing after this manner. Sup pose an obstacle is interposed to hinder the congress of any two bodies A. B, mutually attracting one the other : then if either body, as A, is more attracted towards the other body B, than that other body B is towards the first body A, the obstacle will be more strongly urged by the pressure of the body A than by the pressure of the body B, and therefore will not remain in equilibrio : but the stronger pressure will prevail, and will make the system of the two bodies, together with the obstacle, to move directly OF NATURAL PHILOSOPHY. 93 towards the parts on which B lies ; arid in free spaces, to go forward in infmitiim with a motion perpetually accelerated ; which is absurd and contrary to the first Law. For, by the first Law, the system ought to per severe in its state of rest, or of moving uniformly forward in a right line : and therefore the bodies must equally press the obstacle, and be equally attracted one by the other. I made the experiment on the loadstone and iron. If these, placed apart in proper vessels, are made to float by one another in standing water, neither of them will propel the other ; but, by being equally attracted, they will sustain each other's pressure, and rest at last in an equilibrium. So the gravitation betwixt the earth and its parts is mutual. Let the earth FI be cut by any plane EG into two parts EGF and EGI, and their weights one towards the other will be mutually equal. For if by another plane HK, parallel to the former EG, the greater partFJ EGI is cut into two parts EGKH and HKI. whereof HKI is equal to the part EFG, first cut oft', it is evident that the middle part EGKH, will have no propension by its proper weight towards either side, but will hang as it were, and rest in an equilibrium betwixt both. But the one extreme part HKI will with its whole weight bear upon and press the middle part towards the other extreme part EGF : and therefore the force with which EGI, the sum of the parts HKI and EGKH, tends towards the third part EGF, is equal to the weight of the part HKI, that is, to the weight of the third part EGF. And therefore the weights of the two parts EGI and EGF, one towards the other, are equal, as I was to prove. And in deed if those weights were not equal, the whole earth floating in the non- resisting aether would give way to the greater weight, and, retiring from it, would be carried off in infinitum. And as those bodies are equipollent in the congress and reflexion, whose velocities are reciprocally as their innate forces, so in the use of mechanic instruments those agents are equipollent, and mutually sustain each the contrary pressure of the other, whose velocities, estimated according to the determination of the forces, are reciprocally as the forces. So those weights are of equal force to move the arms of a balance; which during the play of the balance are reciprocally as their velocities upw ards and downwards ; that is, if the ascent or descent is direct, those weights are of equal force, which are reciprocally as the distances of the points at which they are suspended from the axis oi the balance : but if they are turned aside by the interposition of oblique planes, or other ob stacles, and made to ascend or descend obliquely, those bodies will be equipollent, wThich are reciprocally as the heights of their ascent and de scent taken according to the perpendicular ; and that on account of the determination of gravity downwards. 94 THE MATHEMATICAL PRINCIPLES And in like manner in the pully, or in a combination of pullies, the force of a hand drawing the rope directly, which is to the weight, whethel ascending directly or obliquely, as the velocity of the perpendicular ascent of the weight to the velocity of the hand that draws the rope, will sustain the weight. In clocks and such like instruments, made up from a combination of wheels, the contrary forces that promote and impede the motion of the wheels, if they are reciprocally as the velocities of the parts of the wheel on which they are impressed, will mutually sustain the one the other. The force of the screw to press a body is to the force of the hand that turns the handles by which it is moved as the circular velocity of the handle in that part where it is impelled by the hand is to the progressive velocity of the screw towards the pressed body. The forces by which the wedge presses or drives the two parts of the wood it cleaves are to the force of the mallet upon the wedge as the pro- press of the wedge in the direction of the force impressed upon it by the mallet is to the velocity with which the parts of the wood yield to the wedge, in the direction of lines perpendicular to the sides of the wedge. And the like account is to be given of all machines. The power and use of machines consist only in this, that by diminishing the velocity we may augment the force, and the contrary : from whence in all sorts of proper machines, we have the solution of this problem ; 7 move a given weight with a given power, or with a given force to over come any other given resistance. For if machines are so contrived that the velocities of the agent and resistant are reciprocally as their forces, the agent will just sustain the resistant, but with a greater disparity of ve locity will overcome it. So that if the disparity of velocities is so great as to overcome all that resistance which commonly arises either from the attrition of contiguous bodies as they slide by one another, or from the cohesion of continuous bodies that are to be separated, or from the weights of bodies to be raised, the excess of the force remaining, after all those re sistances are overcome, will produce an acceleration of motion proportional thereto, as well in the parts of {he machine as in the resisting body. But to treat of mechanics is not my present business. I was only willing to show by those examples the great extent and certainty of the third Law ot motion. For if we estimate the action of the agent from its force and velocity conjunctly, and likewise the reaction of the impediment conjuncth from the velocities of its several parts, and from the forces of resistance arising from the attrition, cohesion, weight, and acceleration of those parts, the action and reaction 'YL the use of all sorts of machines will b" found always equal to one another. And so far as the action is propagated by the intervening instruments, and at last impressed upon tic resisting body, the ultimate determination of the action will be always contrary to the determination of the reaction. OF NATURAL PHILOSOPHY 95 BOOK I. OF THE MOTION OF BODIES. SECTION I. Of the method of first and last ratios of quantities, by the help wJicreoj we demonstrate the propositions that follow. LEMMA I. Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal. If you deny it, suppose them to be ultimately unequal, and let D be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D ; which is against the supposition, LEMMA II. If in any figure AacE, terminated by the right (f lines A a. AE, and the curve acE, there be in scribed any number of parallelograms Ab, Be, Cd, fyc., comprehended under equal bases AB, BC, CD, ^«c., and the sides, Bb, Cc, Dd, ^c., parallel to one side Aa of the figure ; and the parallelograms aKbl, bLcm, cMdn, §*c., are com pleted. Then if the breadth of those parallelo- \ grams be supposed to be diminisJied, and their X BF C D |; number to be augmented in infinitum : / say, that :he ultimate ratios 'which the inscribed fignre AKbLcMdD, the tin nmscribed figure AalbmcndoE, and en rvilijiear figure AabcdE, will have to one another, are ratios of equality. For the difference of the inscribed and circumscribed figures is the sum of the parallelograms K7, Lw, M//. Do. that is (from the equality of all their bases), the rectangle under one of their bases K6 and the sum of their altitudes Aa, that is, the rectangle ABla. But this rectangle, because M a 96 THE MATHEMATICAL PRINCIPLES [BOOK 1 its breadth AB is supposed diminished • in infinitum, becomes less than any given space. And therefore (by Lem. I) the figures inscribed and circumscribed become ultimately equal one to the other; and much more will the intermediate curvilinear figure be ultimately equal to either* Q.E.D. LEMMA III. The same ultimate ratios are also ratios of equality, when the breadth^ AB, BC, DC, fyc., of the parallelograms are unequal, and are all di minished in infinitum. For suppose AF equal to the greatest breadth, and complete the parallelogram FAaf. This parallelo gram will be greater than the difference of the in scribed and circumscribed figures ; but, because its breadth AF is diminished in infinitum, it will be come less than any given rectangle. Q.E.D. COR. 1. Hence the ultimate sum of those evanes cent parallelograms will in all parts coincide with the curvilinear figure. A BF C D E COR. 2. Much more will the rectilinear figure^comprehendcd under tne chords of the evanescent arcs ab, be, cd, (fee., ultimately coincide with tl.c curvilinear figure. COR. 3. And also the circumscribed rectilinear figure comprehended under the tangents of the same arcs. COR. 4 And therefore these ultimate figures (as to their perimeters acE) are not rectilinear, but curvilinear limi's of rectilinear figures. LEMMA IV. If in two figures AacE, PprT, you inscribe (as before) two ranks of parallelograms, an equal number in each rank, and, when their breadths are diminished in infinitum. the ultimate ratios of the parallelograms in one figure to those in the other, each to each respec tively, are the same; I say, that those two figures AacE, PprT, are to one another in that same ratio. For as the parallelograms in the one are severally to p the parallelograms in the other, so (by composition) is the < sum of all in the one to the sum of all in the other : and so is the one figure to the other; because (by Lem. Ill) the former figure to the former sum, and the latter figure to the latter sum, are both in the ratio of equality. Q.E.D. COR. Hence if two quantities of any kind are any how divided into an equal number of parts, and those A £ SEC. I.] OF NATURAL PHILOSOPHY. 97 parts, when their number is augmented, and their magnitude diminished in infinitum, have a given ratio one to the other, the first to the first, the second to the second, and so on in order, the whole quantities will be one to the other in that same given ratio. For if, in the figures of this Lemma, the parallelograms are taken one to the other in the ratio of the parts, the sum of the parts will always be as the sum of the parallelograms ; and therefore supposing the number of the parallelograms and parts to be aug mented, and their magnitudes diminished in infinitum, those sums will be in the ultimate ratio of the parallelogram in the one figure to the corres pondent parallelogram in the other ; that is (by the supposition), in the ultimate ratio of any part of the one quantity to the correspondent part of the other. LEMMA V. In similar figures, all sorts of homologous sides, whether curvilinear or rectilinear, are proportional ; and the areas are in the duplicate ratio of the homologous sides. LEMMA VI. If any arc ACB, given in position, is snb- _j tended by its chord AB, and in any point A, in the middle of the contiinied curva ture, is touched by a right line AD, pro duced both ways ; then if the points A R and B approach one another and meet, I say, the angle RAT), contained between, the chord and the tangent, will be dimin- ? ished in infinitum, a/id ultimately will vanish. For if that angle does not vanish, the arc ACB will contain with the tangent AD an angle equal to a rectilinear angle ; and therefore the cur vature at the point A will not be continued, which is against the supposi tion. LEMMA VII. The same things being supposed, I say that the ultimate ratio of the arc, chord, and tangent, any one to any other, is the ratio of equality. For while the point B approaches towards the point A, consider always AB and AD as produced to the remote points b and d, and parallel to the secant BD draw bd : and let the arc Acb be always similar to the arc ACB. Then, supposing the points A and B to coincide, the angle dAb will vanish, by the preceding Lemma; and therefore the right lines Ab, Arf (which are always finite), and the intermediate arc Acb, will coincide, and become equal among themselves. Wheref ,re, the right lines AB, AD, 98 THE MATHEMATICAL PRINCIPLES [SEC. I. and the intermediate arc ACB (which are always proportional to the former), will vanish, and ultimately acquire the ratio of equality. Q.E.D. COR. 1. Whence if through B we draw A BP parallel to the tangent, always cutting any right line AF passing through A in F/— i- P, this line BP will be ultimately in the ratio of equality with the evanescent arc ACB ; because, completing the parallelogram APBD, it is always in a ratio of equality with AD. COR. 2. And if through B and A more right lines are drawn, as BE, I5D, AF, AG, cutting the tangent AD and its parallel BP : the ultimate ratio of all the abscissas AD, AE, BF, BG, and of the chord and arc AB, any one to any other, will be the ratio of equality. COR. 3. And therefore in all our reasoning about ultimate ratios, we may freely use any one of those lines for any other. LEMMA VIII. If the right lines AR, BR, with the arc ACB, the chord AB, and the tangent AD, constitute three triangles RAB. RACB, RAD, and the points A and B approach and meet : I say, that the ultimate form oj these evanescent triangles is that of similitude, and their ultimate ratio that of equality. For while the point B approaches towards A the point A, consider always AB, AD, AR, as produced to the remote points b, d, and r, and rbd as drawn parallel to RD, and let the arc Acb be always similar to the arc ACB. Then supposing the points A and B to coincide, the angle bAd will vanish ; and therefore the three triangles rAb, rAcb,rAd ^which are always finite), will coincide, and on that account become both similar and equal. And therefore the triangles RAB. RACB, RAD which are always similar and proportional to these, will ultimately be come both similar and equal among themselves. Q..E.D. COR. And hence in all reasonings about ultimate ratios, we may indif ferently use any one of those triangles for any other. LEMMA IX. If a ngnt line AE. and a curve tine ABC, both given by position, cut each other in a given angle, A ; and to that right line, in another given angle, BD, CE are ordinately applied, meeting the curve in B, C : and the points B and C together approach towards and meet in the point A : / say, that the areas of the triangles ABD, ACE, wilt ultimately be one to the other in the duplicate ratio of the sides. BOOK LI OF NATURAL PHILOSOPHY. For while the points B, C, approach towards the point A, suppose always AD to be produced to the remote points d and . e, so as Ad, Ae may be proportional to AD, AE ; and the ordinates db, ec, to be drawn parallel to the ordinates DB and EC, and meeting AB and AC produced D in b and c. Let the curve A be be similar to the curve A BC, and draw the right line Ag- so as to touch both curves in A, and cut the ordinates DB, EC, db ec, in F, G, J] g. Then, supposing the length Ae to remain the same, let the points B and C meet in the point A ; and the angle cAg vanishing, the curvilinear areas AW, Ace will coincide with the rectilinear areas A/rf, Age ; and therefore (by Lem. V) will be one to the other in the duplicate ratio of the sides Ad, Ae. But the areas ABD, ACE are always proportional to these areas ; and so the sides AD, AE are to these sides. And therefore the areas ABD, ACE are ultimately one to the other in the duplicate ratio of the sides AD, AE. Q.E.D. LEMMA X. The spaces which a bodij describes by any finite force urging it. whether that force is determined and immutable, or is continually augmented or continually diminished, are in the very beginning of the motion one to the other in the duplicate ratio of the times. Let the times be represented by the lines AD, AE, and the velocities generated in those times by the ordinates DB, EC. The spaces described with these velocities will be as the areas ABD, ACE. described by those ordinates, that is, at the very beginning of the motion (by Lem. IX), in the duplicate ratio of the times AD, AE. Q..E.D. COR. 1. And hence one may easily infer, that the errors of bodies des cribing similar parts of similar figures in proportional times, are nearly as the squares of the times in which they are generated ; if so be these errors are generated by any equal forces similarly applied to the bodies, and measured by the distances of the bodies from those places of the sim ilar figures, at which, without the action of those forces, the bodies would have arrived in those proportional times. COR. 2. But the errors that are generated by proportional forces, sim ilarly applied to the bodies at similar parts of the similar figures, are as the forces and the squares of the times conjuiu tly. COR. 3. The same thing is to be understood of any spaces whatsoever described by bodies urged with different forces ; all which, in the very be- g'nning of the motion, are as the forces and the squares of the times conjunctly. 100 THE MATHEMATICAL PRINCIPLES I SEC. 1 COR. 4. And therefore the forces are as the spaces described in the very beginning of the motion directly, and the squares of the times inversely. COR. 5. And the squares of the times are as the spaces described direct ly, und the forces inversely. SCHOLIUM. If in comparing indetermined quantities of different sorts one with another, any one is said to be as any other directly or inversely, the mean ing is, that the former is augmented or diminished in the same ratio with the latter, or with its reciprocal. And if any one is said to be as any other two or more directly or inversely, the meaning is, that the first is aug mented or diminished in the ratio compounded of the ratios in which the others, or the reciprocals of the others, are augmented or diminished. As if A is said to be as B directly, and C directly, and D inversely, the mean ing is, that A is augmented or diminished in the same ratio with B X C X -jj-, that is to say, that A and -£• arc one to the other in a given ratio. LEMMA XL The evanescent subtense of the angle of contact, in all curves which at the point of contact have a finite curvature, is ultimately in the dupli cate rati1) of the subtense of the conterminate arc. CASE 1. Let AB be that arc, AD its tangent, BD the subtense of the angle of contact perpendicular on the tangent, AB the subtense of the arc. Draw BG perpendicular to the subtense AB, and AG to the tan gent AD, meeting in G ; then let the points D, B, and G. approach to the points d, b, and g, and suppose J to be the ultimate intersection of the lines BG, AG, when the points D, B, have come to A. It is evident that the distance GJ may be less than any assignable. But (from the nature of the circles passing through the points A, B, G, A, b, g,) AE2= AG X BD, and A62 = Ag X bd ; and therefore the ratio of AB2 to Ab2 is compounded oi the ratios of AG to Ag, and of Ed to bd. But because GJ may be as sumed of less length than any assignable, the ratio of AG to Ag may be such as to differ from the ratio of equality by less than any assignable difference ; and therefore the ratio of AB2 to Ab2 may be such as to differ from the ratio of BD to bd by less than any assignable difference. There fore, by Lem. I, the ultimate ratio of AB2 to Ab2 is the same with tho ul timate ratio of BD to bd. Q.E.D. CASE 2. Now let BD be inclined to AD in any given an*r1r, and the ultimate ratio of BD to bd will always be the same as before, and there fore the same with the ratio of AB2 to Ab2. Q.E-P BOOK I.] OF NATURAL PHILOSOPHY. 101 CASE 3. And if we suppose the angle D not to be given, but that the right line BD converges to a given point, or is determined by any other condition whatever ; nevertheless the angles D, d, being determined by the same law, will always draw nearer to equality, arid approach nearer to each other than by any assigned difference, and therefore, by Lem. I, will at lust be* equal ; and therefore the lines BD; bd arc in the same ratio to each other as before. Q.E.D. COR. 1. Therefore since the tangents AD, Ad, the arcs AB, Ab, and their sines, BC, be, become ultimately equal to the chords AB, Ab} their squares will ultimately become as the subtenses BD, bd. COR. 2. Their squares are also ultimately as the versed sines of the arcs, bisecting the chords, and converging to a given point. For those versed sines are as the subtenses BD, bd. COR. 3. And therefore the versed sine is in the duplicate ratio of the time in which a body will describe the arc with a given velocity. COR. 4. The rectilinear triangles ADB, Adb are ultimately in the triplicate ratio of the sides AD, Ad, c and in a sesquiplicate ratio of the sides DB, db ; as being in the ratio compounded of the sides AD to DB, and of Ad to db. So also the triangles ABC, Abe are ultimately in the triplicate ratio of the sides BC, be. What I call the sesquiplicate ratio is the subduplicate of the triplicate, as being compounded of the simple and subduplicate ratio. j COR. 5. And because DB, db are ultimately paral- g lei and in the duplicate ratio of the lines AD, Ad, the ultimate curvilinear areas ADB, Adb will be (by the nature of the para bola) two thirds of the rectilinear triangles ADB, Adb and the segments AB, Ab will be one third of the same triangles. And thence those areas and those segments will be in the triplicite ratio as well of the tangents AD, Ad, as of the chords and arcs AB, AB. SCHOLIUM. But we have all along supposed the angle of contact to be neither infi nitely greater nor infinitely less than the angles of contact made by cir cles and their tangents ; that is, that the curvature at the point A is neither infinitely small nor infinitely great, or that the interval AJ is of a finite mag nitude. For DB may be taken as AD3 : in which case no circle can be drawn through the point A, between the tangent AD and the curve AB, and therefore the angle of contact will be infinitely less than those of circles. And by a like reasoning, if DB be made successfully as AD4, AD5, AD8, AD7, etc., we shall have a series of angles of contact, proceeding in itifini- tum, wherein every succeeding term is infinitely less than the pre- 102 THE MATHEMATICAL PRINCIPLES [BOOK 1 ceding. And if DB be made successively as AD2, AD|, AD^, AD], AD| AD7, &c., we shall have another infinite series of angles of contact, the first of which is of the same sort with those of circles, the second infinitely greater, and every succeeding one infinitely greater than the preceding. But between any two of these angles another series of intermediate angles of contact may be interposed, proceeding both ways in infinitum. wherein every succeeding angle shall be infinitely greater or infinitely less than the preceding. As if between the terms AD2 and AD3 there were interposed the series AD'f, ADy, AD49, AD|, AD?, AD|, AD^1, AD^, AD^7, &c. And again, between any two angles of this series, a new series of intermediate angles may be interposed, differing from one another by infinite intervals. Nor is nature confined to any bounds. Those things which have been demonstrated of curve lines, and the euperfices which they comprehend, may be easily applied to the curve su- perfices and contents of solids. These Lemmas are premised to avoid the tediousness of deducing perplexed demonstrations ad absurdnm, according to the method of the ancient geometers. For demonstrations are more contracted by the method of indivisibles : but because the hypothesis of indivisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the follow ing propositions to the first and last sums and ratios of nascent and evane scent quantities, that is, to the limits of those sums and ratios ; and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is performed as by the method of indivisibles ; and now those principles being demonstrated, we may use them with more safety. Therefore if hereafter I should happen to consider quantities as made up of particles, or should use little curve lines for right ones, I would not be un- (lerstood to mean indivisibles, but evanescent divisible quantities : not the sums and ratios of determinate parts, but always the limits of sums and ratios ; and that the force of such demonstrations always depends on the method laid down in the foregoing Lemmas. Perhaps it may be objected, that there is no ultimate proportion, of evanescent quantities ; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alledged, that a body arriving at a cer tain place, and there stopping has no ultimate velocity : because the velo city, before the body comes to the place, is not its ultimate velocity ; when it has arrived, is none i ut the answer is easy; for by the ultimate ve locity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives ; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ra tio of evanescent quantities is to Le understood the ratio of the ijuantitiea SEC. II.] OF NATURAL PHILOSOPHY. 103 not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the ve locity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and pro portions that begin and cease to be. And since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geometrical we may be allowed to use in determining and de monstrating any other thing that is likewise geometrical. It may also be objected, that if the ultimate ratios of evanescent quan tities are given, their ultimate magnitudes will be also given : and so all quantities will consist of indivisibles, which is contrary to what Euclid has demonstrated concerning incommensurables, in the 10th Book of his Elements. But this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge ; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in wfinitum. This thing will appear more evident in quantities infinitely great. If two quantities, whose dif ference is given, be augmented in infin&um, the ultimate ratio of these quantities will be given, to wit, the ratio of equality ; but it does not from thence follow, that the ultimate or greatest quantities themselves, whose ratio that is, will be given. Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least, or evanescent, or ultimate, you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be al ways diminished without end. SECTION II. Of the Invention of Centripetal Forces. PROPOSITION I. THEOREM 1. The areas, which revolving bodies describe by radii drawn to an ^mmo- vable centra of force do lie in tJ:e same immovable planes, and are pro- portional to the times in which they are described. For suppose the time to be divided into equal parts, and in the first part of that time let the body by its innate force describe the right line AB In the second part of that time, the same would (by Law I.), if not hindered, proceel directly to c, alo ILJ; the line Be equal to AB ; so that by the radii AS, BS, cS, draw. i to the centre, the equal areas ASB, BSc, would be de- 104 THE MATHEMATICAL PRINCIPLES [BOOK I scribed. But when the body is arrived at B, suppose that a centripetal force acts at once with a great im pulse, and, turning aside the body from the right line Be, compels it afterwards to con tinue its motion along the right line BC. Draw cC parallel to BS meeting BC in C ; and at the end of the second part of the time, the body (by Cor. I. of the Laws) will be found in C, in the same plane with the triangle A SB. Join SC, and, because s SB and Cc are parallel, the triangle SBC will be equal to the triangle SBc, and therefore also to the triangle SAB. By the like argument, if the centripetal force acts successively in C, D, E. &c., and makes the body, in each single particle of time, to describe the right lines CD, DE, EF7 &c., they will all lie in the same plane : and the triangle SCD will be equal to the triangle SBC, and SDE to SCD, and SEF to SDE. And therefore, in equal times, equal areas are described in one immovable plane : and, by composition, any sums SADS, SAFS, of those areas, are one to the other as the times in which they are described. Now let the number of those triangles be augmented, and their breadth diminished in wjinitum ; and (by Cor. 4, Lem. III.) their ultimate perimeter ADF will be a curve line : and therefore the centripetal force, by which the body is perpetually drawn back from the tangent of this curve, will act continually ; and any described areas SADS, SAFS, which are always proportional to the times of de scription, will, in this case also, be proportional to those times. Q.E.D. COR. 1. The velocity of a body attracted towards an immovable centre, in spaces void of resistance, is reciprocally as the perpendicular let fall from that centre on the right line that touches the orbit. For the veloci ties in those places A, B, C, D, E. are as the bases AB, BC, CD, DE, EF. of equal triangles ; and these bases are reciprocally as the perpendiculars let fall upon them. COR. 2. If the chords AB, BC of two arcs, successively described in equal times by the same body, in spaces void of resistance, are completed into a parallelogram ABCV, and the diagonal BV of this parallelogram; in the position which it ultimately acquires when those arcs are diminished in irifinitum, is produced both ways, it will pass through the centre of force. COR. 3. If the chords AB, BC, and DE, EF, cf arcs described in equal SEC. II.] OF NATURAL PHILOSOPHY. 105 times, in spaces void of resistance, are completed into the parallelograms ABCV, DEFZ : the forces in B and E are one to the other in the ulti mate ratio of the diagonals BV, EZ, when those arcs are diminished in infinitum. For the motions BC and EF of the body (by Cor. 1 of the Laws) are compounded of the motions Be, BV, and E/", EZ : but BV and EZ, which are equal to Cc and F/, in the demonstration of this Proposi tion, were generated by the impulses of the centripetal force in B and E; and are therefore proportional to those impulses. COR. 4. The forces by which bodies, in spaces void of resistance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are one to another as the versed sines of arcs described in equal times ; which versed sines tend to the centre of force, and bisect the chords when those arcs are diminished to infinity. For such versed sines are the halves of the diagonals mentioned in Cor. 3. COR. 5. And therefore those forces are to the force of gravity as the said versed sines to the versed sines perpendicular to the horizon of those para bolic arcs which projectiles describe in the same time. COR. 6. And the same things do all hold good (by Cor. 5 of the Laws), when the planes in which the bodies are moved, together with the centres of force which are placed in those planes, are not at rest, but move uni formly forward in right lines. PROPOSITION II. THEOREM II. Every body that moves in any curve line described in a plane, and by a radius, drawn to a point either immovable, or moving forward with an uniform rectilinear motion, describes about that point areas propor tional to the times, is urged by a centripetal force directed to that point CASE. 1. For every body that moves in a curve line, is (by Law 1) turned aside from its rectilinear course by the action of some force that impels it. And that force by which the body is turned off from its rectilinear course, and is made to describe, in equal times, the equal least triangles SAB, SBC, SCD, &c., about the immovable point S (by Prop. XL. Book 1, Elem. and Law II), acts in the place B, according to the direction of a line par- 1U6 THE MATHEMATICAL PRINCIPLES [BOOK f. allel K cC. that is, in the direction of the line BS. and in the place C, accordii g to the direction of a line parallel to dD, that is, in the direction of the line CS, (fee.; and therefore acts always in the direction of lines tending to the immovable point S. Q.E.I). CASE. 2. And (by Cor. 5 of the Laws) it is indifferent whether the su- perfices in which a body describes a curvilinear figure be quiescent, or moves together with the body, the figure described, and its point S, uniformly forward in right lines. COR. 1. In non-resisting spaces or mediums, if the areas are not propor tional to the times, the forces are not directed to the point in which the radii meet ; but deviate therefrom in. consequently or towards the parts to which the motion is directed, if the description of the areas is accelerated ; but in antecedentia, if retarded. COR. 2. And even in resisting mediums, if the description of the areas is accelerated, the directions of the forces deviate from the point in which the radii meet, towards the parts to which the motion tends. SCHOLIUM. A body may be urged by a centripetal force compounded of several forces ; in which case the meaning of the Proposition is, that the force which results out of all tends to the point S. But if any force acts per petually in the direction of lines perpendicular to the described surface, this force will make the body to deviate from the plane of its motion : but will neither augment nor diminish the quantity of the described surface and is therefore to be neglected in the composition of forces. PROPOSITION III. THEOREM III. Every body, that by a radius drawn to the centre of another body, how soever moved, describes areas about that centre proportional to iJie times, is urged by a force compounded out of the centripetal force Bending fo that other body, and of all the accelerative force by which that other body is impelled. Let L represent the one, and T the other body ; and (by Cor. 0 of the Laws) if both bodies are urged in the direction of parallel lines, by a ne T force equal and contrary to that by which the second body T is tinned, the first body L will go on to describe about the other body T the same areas as before : but the force by which that other body T was urged will be now destroyed by an equal and contrary force; and therefore (by Law I.) that other body T, now left to itself, will either rest, or move uniformly forward in a right line : and the first body L impelled by the difference of the forces, that is, by the force remaining, will go on to describe about the other body T areas proportional to the times. And therefore (by Theor. II.) the difference ;f the forces is directed to the other body T as its centre. Q.E.D SEC. IL] OF NATURAL PHILOSOPHY. 107 Co.*. 1. Hence if the one body L, by a radius drawn to the other body T, describes areas proportional to the times ; and from the whole force, by which the firr.t body L is urged (whether that force is simple, or, according to Cor. 2 of the Laws, compounded out of several forces), we subduct (by the same Cor.) that whole accelerative force by which the other body is urged ; the who_e remaining force by which the first body is urged will tend to the ( ther body T, as its centre. COR. 2. And, if these areas are proportional to the times nearly, the re maining force will tend to the other body T nearly. COR. 3. And vice versa, if the remaining force tends nearly to the other body T, those areas will be nearly proportional to the times. COR. 4. If the body L, by a radius drawn to the other body T, describes areas, which, compared with the times, are very unequal ; and that other body T be either at rest, or moves uniformly forward in a right line : the action of the centripetal force tending to that other body T is either none at all, or it is mixed and compounded with very powerful actions of other forces : and the whole force compounded of them all, if they are many, is directed to another (immovable or moveaJble) centre. The same thing ob tains, when the other body is moved by any motion whatsoever ; provided that centripetal force is taken, wrhich remains after subducting that whole force acting upon that other body T. SCHOLIUM. Because the equable description of areas indicates that a centre is re spected by that force with which the body is most affected, and by which it is drawn back from its rectilinear motion, and retained in its orbit ; why may we not be allowed, in the following discourse, to use the equable de scription of areas as an indication of a centre, about which all circular motion is performed in free spaces ? PROPOSITION IV. THEOREM IV. The centripetal forces of bodies, which by equable 'motions describe differ ent circles, tend to the centres of the same circles ; and are one to tJie other as the squares of t/ie arcs described in equal times applied to the radii of the circles. These forces tend to the centres of the circles (by Prop. II., and Cor. 2, Prop. L), and are one to another as the versed sines of the least arcs de scribed in equal times (by Cor. 4, Prop. I.) ; that is, as the squares of the same arcs applied to the diameters of the circles (by Lem. VII.) ; and there fore since those arcs are as arcs described in any equal times, and the dia- me ers ace as the radii, the forces will be as the squares of any arcs de- scr bed in the same time applied to the radii of the circles. Q.E.D. ^OR. 1. Therefore, since those arcs are as the velocities of the bodies. I OS THE MATHEMATICAL PRINCIPLES [BOOK . the centripetal forces are in a ratio compounded of the duplicate ra'jio of the velocities directly, and of the simple ratio of the radii inversely. COR. 2. And since the periodic times are in a ratio compounded of the ratio of the radii directly, and the ratio of the velocities inversely, the cen tripetal forces, are in a ratio compounded of the ratio of the radii directly, and the duplicate ratio of the periodic times inversely. COR, 3. Whence if the periodic times are equal, and the velocities therefore as the radii, the centripetal forces will be also as the radii ; and tke contrary. COR. 4. If the periodic times and the velocities are both in the subdu- plicate ratio of the radii, the centripetal forces will be equal among them selves ; and the contrary. COR. 5. If the periodic times are as the radii, and therefore the veloci ties equal, the centripetal forces will be reciprocally as the radii ; and the contrary. COR. 6. If the periodic times are in the sesquiplicate ratio of the radii, and therefore the velocities reciprocally in the subduplicate ratio of the radii, the centripetal forces will be in the duplicate ratio of the radii in versely : and the contrary. COR. 7. And universally, if the periodic time is as any power Rn of the radius R, and therefore the velocity reciprocally as the power Rn ] of the radius, the centripetal force will be reciprocally as the power R2n 1 of the radius; and the contrary. COR. 8. The same things all hold concerning the times, the velocities, and forces by which bodies describe the similar parts of any similar figures that have their centres in a similar position with those figures ; as appears by applying the demonstration of the preceding cases to those. And the application is easy, by only substituting the equable description of areas in the place of equable motion, and using the distances of the bodies from the centres instead of the radii. COR. 9. From the same demonstration it likewise follows, that the arc which a body, uniformly revolving in a circle by means of a given centri petal force, describes in any time, is a mean proportional between the diameter of the circle, and the space which the same body falling by the same given force would descend through in the same given time. SCHOLIUM. The case of the 6th Corollary obtains in the celestial bodies (as Sir Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed) ; and therefore in what follows, I intend to treat more at large of those things which relate to centripetal force decreasing in a duplicate ratio of the distances from the centres. Moreover, by means of the preceding Proposition and its Corollaries, we SEC. II.] OF NATURAL PHILOSOPHY. 109 may discover the proportion of a centripetal force to any other known force, such as that of gravity. For if a body by means of its gravity re volves in a circle concentric to the earth, this gravity is the centripetal force of that body. But from the descent of heavy bodies, the time of one entire revolution, as well as the arc described in any given time, is given (by Cor. 9 of this Prop.). And by such propositions, Mr. Huygens, in his excellent book De Horologio Oscillatorio, has compared the force of gravity with the centrifugal forces of revolving bodies. The preceding Proposition may be likewise demonstrated after this manner. In any circle suppose a polygon to be inscribed of any number of sides. And if a body, moved with a given velocity along the sides of the polygon, is reflected from the circle at the several angular points, the force, with which at every reflection it strikes the circle, will be as its velocity : and therefore the sum of the forces, in a given time, will be as that ve locity and the number of reflections conjunctly ; that is (if the species of the polygon be given), as the length described in that given time, and in creased or diminished in the ratio of the same length to the radius of the circle ; that is, as the square of that length applied to the radius ; and therefore the polygon, by having its sides diminished in'inftnitum, coin cides with the circle, as the square of the arc described in a given time ap plied to the radius. This is the centrifugal force, with which the body impels the circle ; and to which the contrary force, wherewith the circle continually repels the body towards the centre, is equal. PROPOSITION V. PROBLEM I. There being given, in any places, the velocity with which a body de scribes a given figure, by means of forces directed to some common centre : to find that centre. Let the three right lines PT, TQV, VR touch the figure described in as many points, P, Q, R, and meet in T and V. On the tan gents erect the perpendiculars PA, QB, RC, reciprocally proportional to the velocities of the body in the points P, Q, R, from which the perpendiculars were raised ; that is, so that PA may be to QB as the velocity in Q to the velocity in P, and QB to RC as the velocity in R to the velocity in Q. Through the ends A, B, C, of the perpendiculars draw AD, DBE, EC, at right angles, meeting in D and E : and the right lines TD, VE produced, will meet in S, the centre re quired. For the perpendiculars let fall from the centre S on the tangents PT. QT. are reciprocally as the velocities of the bodies in the points P and Q 110 THE MATHEMATICAL PRINCIPLES [BOOK 1 (by Cor. 1, Prop. I.), and therefore, by construction, as the perpendiculars AP, BQ, directly ; that is, as the perpendiculars let fall from the point D on the tangents. Whence it is easy to infer that the points S, D, T, are in one right line. And by the like argument the points S, E, V are also in one right line ; and therefore the centre S is in the point where the right lines TD; YE meet. Q.E.D. PROPOSITION VL THEOREM V. In a space void of resistance, if a body revolves in any orbit about an im movable centre, and in the least time describes any arc just then, na scent ; and the versed sine of that arc is supposed to be drawn bisect ing the chord, and produced passing through the centre of force: the centripetal force in the middle of the arc will be as the versed sine di rectly and the square of the time inversely. For the versed sine in a given time is as the force (by Cor. 4, Prop. 1) ; and augmenting the time in any ratio, because the arc will be augmented in the same ratio, the versed sine will be augmented in the duplicate of that ratio (by Cor. 2 and 3, Lem. XL), and therefore is as the force and the square of the time. Subduct on both sides the duplicate ratio of the time, and the force will be as the versed sine directly, arid the square of the time inversely. Q.E.D. And the same thing may also be easily demonstrated by Corol. 4? T,em. X. COR. 1. If a body P revolving about the centre S describes a curve line APQ,, which a right line ZPR touches in any point P ; and from any other point Q, of the curve, QJl is drawn parallel to the distance SP, meeting the tangent in R ; and QT is drawn perpen- (licular to the distance SP ; the centripetal force will be reciprocally as the sp2 x Q/r2 solid - — :— , if the solid be taken of that magnitude which it ulti- mately acquires when the points P and Q, coincide. For Q,R is equal to the versed sine of double the arc QP, whose middle is P : and double the triangle SQP, or SP X Q,T is proportional to the time in which that double arc is described ; and therefore may be used for the exponent of the time. COR. 2. By a like reasoning, the centripetal force is reciprocally as the SY2 X QJP2 solid - 7^5 - ; if SY is a perpendicular from the centre of force on PR the tangent of the orbit. For the rectangles SY X QP and SP X Q,T are equal. SEC. II.] OF NATURAL PHILOSOPHY. Ill COR. 3. If the orbit is cither a circle, or touches or cuts a circle c< ncen- trically, that is, contains with a circle the least angle of contact or sec tion, having the same curvature rnd the same radius of curvature at the point P : and if PV be a chord of this circle, drawn from the body through the centre of force ; the centripetal force will be reciprocally as the solid QP2 SY2 X PV. For PV is - . COR. 4. The same things being supposed, the centripetal force is as the square of the velocity directly, and that chord inversely. For the velocity is reciprocally as the perpendicular SY, by Cor. 1. Prop. I. COR. 5. Hence if any curvilinear figure APQ, is given, and therein a point S is also given, to which a centripetal force is perpetually directed. that law of centripetal force may be found, by which the body P will bcj continually drawn back from a rectilinear course, and. being detained in the perimeter of that figure, will describe the same by a perpetual revolu- SP2 x QT2 tion. That is, we are to find, by computation, either the solid ---- - or the solid SY2 X PV, reciprocally proportional to this force. Example: of this we shall give in the following Problems. PROPOSITION VII. PROBLEM II. Tf a body revolves in the circumference of a circle; it is proposed to finii the law of centripetal force directed to any given, point. Let VQPA be the circumference of the circle ; S the given point to which as to a centre the force tends : P the body mov ing in the circumference ; Q the next place into which it is to move; and PRZ the tangent of the circle at the preceding place. Through the point S draw the v chord PV, and the diameter VA of the circle : join AP, and draw Q,T perpen dicular to SP, which produced, may meet the tangent PR in Z ; and lastly, through the point Q, draw LR parallel to SP, meeting the circle" in L, and the tangent PZ in R. And, because of the similar triangles ZQR, ZTP. VPA, we shall have RP2, that is. QRL to QT2 as AV2 to PV2. And QRlj x PV2 SI3-' therefore '- —TS -- is equal to QT2. Multiply those equals by -'. and the points P and Q, coinciding, for RL write PV ; then we shall have SP-' X PV5 SP2 x QT2 — • And therefore fl»r Cor 1 and 5. Prop. VI.) 112 THE MATHEMATICAL PRINCIPLES [BOOK I, SP2 X PV3 the centripetal force is reciprocally as - — ry^~ — J that is (because AV2 ia given), reciprocally as the square of the distance or altitude SP, and the 3ube of the chord PV conjunctly. Q.E.L The same otherwise. On the tangent PR produced let fall the perpendicular SY ; and (be cause of the similar triangles SYP, VPA), we shall have AV to PV as SP SP X PV SP2 >< PV3 to SY, and therefore -- ^~— - = SY, and - — ^- = SY2 X PV. A V A V And therefore (by Corol. 3 and 5, Prop. VI), the centripetal force is recip- SP2 X PV3 rocally as - ~~ry¥~~~ I *na* *s (because AV is given), reciprocally as SP" X PV3. Q.E.I. Con. 1. Hence if the given point S, to which the centripetal force al ways tends, is placed in the circumference of the circle, as at V, the cen tripetal force will be reciprocally as the quadrato-cube (or fifth power) of the altitude SP. COR. 2. The force by which the body P in the circle APTV revolves about the centre of force S is to the force by which the same body P may re volve in the same circle, and in the same periodic time, about any other centre of force R, as RP2 X SP to the cube of the right line SG, which, from the first centre of force S is drawn parallel to the distance PR of the body from the second centre of force R, meeting the tangent PG of the orbit in G. For by the construction of this Proposition, the former force is to the latter as RP2 X PT3 to SP2 X PV3; that is, as SP3 X PV3 SP X RP2 to -- p™ — ; or (because of the similar triangles PSG, TPV) to SGS. COR. 3. The force by which the body P in any orbit revolves about the centre of force S, is to the force by which the same body may revolve in the same orbit, and the same periodic time, about any other centre of force R. as the solid SP X RP2, contained under the distance of the body from the first centre of force S, and the square of its distance from the sec ond centre of force R, to the cube of the right line SG, drawn from the first centre of the force S, parallel to the distance RP of the body from fch*3 second centre of force R, meeting the tangent PG of the orbit in G. For the force in this orbit at any point P is the same as in a circle of the same curvature. SJSG. IL] OF NATURAL PHILOSOPHY. 113 PROPOSITION VIII. PROBLEM III. If a body mi ues in the semi-circuwferencePQA: it is proposed to find the law of the centripetal force tending to a point S, so remote, that all the lines PS. RS drawn thereto, may be taken for parallels. From C, the centre of the semi-circle, let the semi-diameter CA he drawn, cutting the parallels at right angles in M and N, and join CP. Because of the similar triangles CPM, PZT, and RZQ, we shall have CP2 to PM2 as PR2 to QT2; and, from the na ture of the circle, PR2 is equal to the rect angle QR X RN + QN, or, the points P, Q coinciding, to the rectangle QR x 2PM. Therefore CP2 is to PM2 as QR X 2PM to QT2; and QT2 2PM3 QT2 X SP2 2PM3 X SP2 therefore (by QR Corol. 8PM3 X SP2 , and QR And 1 and 5, Prop. VI.), the centripetal force is reciprocally as 2SP2 . that is (neglecting the given ratio -ppr)> reciprocally as PM3. Q.E.L And the same thing is likewise easily inferred from the preceding Pro position. SCHOLIUM. And by a like reasoning, a body will be moved in an ellipsis, or even ia an hyperbola, or parabola, by a centripetal force which is reciprocally ae the cube of the ordinate directed to an infinitely remote centre of force. PROPOSITION IX. PROBLEM IV. If a body revolves in a spiral PQS, cutting all the radii SP, SQ, fyc., in a given angle; it is proposed to find thelaio of the centripetal force tending to tJie centre of that spiral. Suppose the inde finitely small angle AY PSQ to be given ; be cause, then, all the angles are given, the figure SPRQT will , _ be given in specie. v QT Q,T2 Therefore the ratio -7^- is also given, and „„ is as QT, that is (be lot IX QK cause the figure is given in specie), as SP. But if the angle PSQ is any way changed, the right line QR, subtending the angle of contact QPU tU THE MATHEMATICAL PRINCIPLES [BOOK J (by Lemma XI) will be changed in the duplicate ratio of PR or QT QT2 Therefore the ratio ~TVD~ remains the same as before, that is, as SP. And QT2 x SP2 — -^ is as SP3, and therefore (by Corol. 1 and 5, Prop. YI) the centripetal force is reciprocally as the cube of the distance SP. Q.E.I. The same otherwise. The perpendicular SY let fall upon the tangent, and the chord PY of the circle concentrically cutting the spiral, are in given ratios to the height SP ; and therefore SP3 is as SY2 X PY, that is (by Corol. 3 and 5, Prop. YI) reciprocally as the centripetal force. LEMMA XII. All parallelograms circumscribed about any conjugate diameters of a given ellipsis or hyperbola are equal among themselves. This is demonstrated by the writers on the conic sections. PROPOSITION X. PROBLEM Y. If a body revolves in an ellipsis ; it is proposed to find the law of thi centripetal force tending to the centre of the ellipsis. Suppose CA, CB to be semi-axes of the ellipsis; GP, DK, con jugate diameters ; PF, Q,T perpendiculars to those diameters; Qvan ^rdinate to the diame ter GP ; and if the parallelogram QvPR be completed, then (by the properties of the jonic sections) the rec- langle PvG will be to Qv2 as PC2 to CD2; and (because of the similar triangles Q^T, PCF), Qi>2 to QT2 as PC2 to PF2 ; and, by com position, the ratio of PvG to QT2 is compounded of the ratio of PC2 1< QT2 CD2, and of the ratio of PC2 to PF2, that is, vG to -p— as PC; to_92L^_P_]^_. Put QR for Pr, and (by Lem. XII) BC X CA for CD K PF ; also (the points P and Q coinciding) 2PC for rG; and multiply- SEC. II.] OF NATURAL PHILOSOPHY. 115 QT2 x PC2 ing the extremes and means together, we shall have rfo~ equal to 2BC2 X CA2 — pp — — . Therefore (by Cor. 5, Prop. VI), the centripetal force is 2BC2 X CA2 reciprocally as — — ry~ ; that is (because 2I3C2 X CA2 is given), re ciprocally as-r^v; that is, directly as the distance PC. QEI. I O TJie same otherwise. [n the right line PG on the other side of the point T, take the point u so that Tu may be equal to TV ; then take uV, such as shall be to v G as DC2 to PC2. And because Qr9 is to PvG as DC'2 to PC2 (by the conic sections), we shall have Qv2 •-= Pi' X «V. Add the rectangle n.Pv to both sides, and the square of the chord of the arc PQ, will be equal to the rect angle VPv ; and therefore a circle which touches the conic section in P, and passes through the point Q,, will pass also through the point V. Now let the points P and Q, meet, and the ratio of nV to rG, which is the same with the ratio of DC2 to PC2, will become the ratio of PV to PG, or PV 2DC2 to 2PC : and therefore PY will be equal to „„ — . And therefore the force by which the body P revolves in the ellipsis will be reciprocally as 2 DC2 — ry— X PF2 (by Cor. 3, Prop. VI) ; that is (because 2DC2 X PF2 is I O given) directly as PC. Q.E.I. COR. 1. And therefore the force is as the distance of the body from the centre of the ellipsis ; and, vice versa, if the force is as the distance, the body will move in an ellipsis whose centre coincides with the centre of force, or perhaps in a circle into which the ellipsis may degenerate. COR. 2. And the periodic times of the revolutions made in all ellipses whatsoever about the same centre will be equal. For those times in sim ilar ellipses will be equal (by Corol. 3 and S, Prop. IV) ; but in ellipses that have their greater axis common, they are one to another as the whole areas of the ellipses directly, and the parts of the areas described in the same time inversely: that is, as the lesser axes directly, and the velocities of the bodies in their principal vertices inversely ; :hat is, as those lesser axes dirtily, and the ordinates to the same point %f the common axes in versely ; and therefore (because of the equality of the direct and inverse ratios) in the ratio of equality. SCHOLIUM. If the ellipsis, by having its centre removed to an infinite distance, de generates into a parabola, the body will move in tin's parabola ; and the 116 THE MATHEMATICAL PRINCIPLES [BOOK I force, now tending to a centre infinitely remote, will become equable. Which is Galileo s theorem. And if the parabolic section of the cone (by changing the inclination of the cutting plane to the cone) degenerates into an hyperbola, the body will move in the perimeter of this hyperbola, hav ing its centripetal force changed into a centrifugal force. And in like manner as in the circle, or in the ellipsis, if the forces are directed to the centre of the figure placed in the abscissa, those forces by increasing or di minishing the ordinates in any given ratio, or even by changing the angle of the inclination of the ordinates to the abscissa, are always augmented or diminished in the ratio of the distances from the centre ; provided the periodic times remain equal ; so also in all figures whatsoever, if the ordi- nates are augmented or diminished in any given ratio, or their inclination is any way changed, the periodic time remaining the same, the forces di rected to any centre placed in the abscissa are in the several ordinatee augmented or diminished in the ratio of the distances from the centre SECTION III. Of the motion of bodies in eccentric conic sections. PROPOSITION XL PROBLEM VI. If a body revolves in an ellipsis ; it is required to find the law of the centripetal force tending to the focus of the ellipsis. Let S be the focus of the ellipsis. Draw SP cutting the diame ter DK of the ellipsis in E, and the ordinate Qv in x ; and com plete the parallelogram d.rPR, It is evident that EP is equal to the greater semi-axis AC : for drawing HI froln the other focus H of the ellipsis parallel to EC, because CS, CH are equal, ES, El will be also equal ; so that EP is the half sum of PS, PI, that is (because of the parallels HI, PR, and the equal angles IPR, HPZ), of PS, PH, which taken together are equal to the whole axis 2AC. Draw QT perpendicu lar to SP, and putting L for the princi al latus rectum of the ellipsis (or for III. OF NATURAL PHILOSOPHY. 117 L X ^R t0 L X Py aS ^R t0 PV that 1S> US PE or AC to PC ; and L X Pv to GvP as L to Gy ; and GvP to Qi>2 as to CD- ; and by (Corol. 2, Lem. VII) the points Q, and P coinciding, Qv* is to Q,r- in the ratio of equality ; and Q,.r2 or Qv2 is to Q,T2 as EP2 to PF2, that is, as CA2 to PF2, or (by Lem. XII) as CD'2 to CB2. And com pounding all those ratios together, we shall have L X QR to Q,T2 as AC X L X PC2 X CD2, or 2CB2 X PC2 X CD2 to PC X Gv X CD2 X CB2, or as 2PC to Gv. But the points Q and P coinciding, 2PC and Gr are equal. And therefore the quantities L X QR and Q,T2, proportional SP2 to these, will be also equal. Let those equals be drawn into-p^B"? and L SP2 X QT2 X SP2 will become equal to -- ^p — — . And therefore (by Corol. 1 and 5, Prop. VI) the centripetal force is reciprocally as L X SP2, that is, re ciprocally in the duplicate ratio of the distance SP. Q.E.I. The same otherwise. Since the force tending to the centre of the ellipsis, by which the body P may revolve in that ellipsis, is (by Corol. 1, Prop. X.) as the distance CP of the body from the centre C of the ellipsis ; let CE be drawn paral lel to the tangent PR of the ellipsis : and the force by which the same body P may revolve about any other point S of the ellipsis, if CE and PS in- PE3 tersect in E, will be as ^T3, (by Cor. 3, Prop. VII.) ; that is, if the point S is the focus of the ellipsis, and therefore PE be given as SP2 recipro cally. Q.E.I. With the same brevity with which we reduced the fifth Problem to the parabola, and hyperbola, we might do the like here : but because of the dignity of the Problem and its use in what follows, I shall confirm the other cases by particular demonstrations. PROPOSITION XII. PROBLEM VII. Suppose a body to move in an hyperbola ; it is required to find lite law of the centripetal force tending to the focus of that figure. Let CA, CB be the semi-axes of the hyperbola ; PG, KD other con jugate diameters ; PF a perpendicular to the diameter KD ; and Qv an ordinate to the diameter GP. Draw SP cutting the diameter DK in E, and the ordinate Qv in x, and complete the parallelogram QRP.r. It is evident that EP is equal to the semi-transverse axis AC ; for drawing HE, from the other focus H of the hyperbola, parallel to EC, because CS, TH are equal, ES El will be also equal ; so that EP is the half difference J1S THE MATHEMATICAL PRINCIPLES [Book I .of PS, PI; that is (be cause of the parallels IH, PR, and the equal angles IPR, HPZ), of PS, PH, the difference of which is equal to the whole axis 2AC. Draw Q,T perpen dicular to SP; and put ting L for the principal latus rectum of the hy perbola (that is, for 2BC2\ .... -Tp- ) 7 we shall have L X QR to L X Pv as QR to Pv, or Px to Pv, that is (because of the similar tri angles Pxv, PEC), as PE to PC, or AC to PC. And L X Pv will be to Gv X Pv as L to Gv; and (by the properties of the conic sections) the rec tangle G?'P is to Q,v2 as PC2 to CD2 ; and by (Cor. 2, Lem. VII.), Qv2 to Qa* the points Q and P coinciding, becomes a ratio of equality ; and Q,.r2 or Qv2 is to Q,T2 as EP2 to PF2, that is, as CA2 to PF2, or (by Lem. XII.) as CD2 to CB2 : and, compounding all those ratios together, we shall have L X Q,R to Q,T2 as AC X L X PC2 X CD2, or 2CB2 X PC2 X CD2 to PC X Gv X CD2 X CB2, or as 2PC to Gv. But the points P and Q, coinciding. 2PC and Gv are equal. And therefore the quantities L X Q,R arid Q.T2, propor tional to them, will be also equal. Let those equals be drawn into SP2 sp2 x o/r2 ^, and we shall have L X SP2 equal to ^^ . And therefore (by Cor. 1. and 5, Prop. VI.) the centripetal force is reciprocally as L X SP'2. 'hat is, reciprocally in the duplicate ratio of the distance SP. Q,.E.I. TJie same otherwise. Find out the force tending from the centre C of the hype rbola. This will be proportional to the distance CP. But from thence (by Cor. 3, Prop. PE3 VII.) the force tending to the focus S will be as -^-^ th; (t is, because PE is given reciprocally as SP-. Q,.E.I. SEC. III.] OF NATURAL PHILOSOPHY. 119 And the same way may it be demonstrated, that the body having its cen tripetal changed into a centrifugal force, will move in the conjugate hy perbola. LEMMA XIII. The latus rectum of a parabola belonging to any vertex is quadruple the distance of that vertex from the focus of thejigurc. This is demonstrated by the writers on the conic sections. LEMMA XIV. Tlie perpendicular, let fall from the focus of a parabola on its tangent, is a mean proportional between the. distances of the focus from the poini of contact, and from the principal vertex of the figure. For, let AP be the parabola, S its focus, A its principal vertex, P the point of contact, PO an ordinate to the principal diameter. PM the tangent meeting the principal diameter in M. and SN the perpendicular from the fo-~ M A s o cus on the tangent : join AN, and because of the equal lines MS and SP, MN and NP, MA and AC, the right lines AN, OP, will be parallel ; and thence the triangle SAN will be right-angled at A, and similar to the equal triangles SNM, SNP j therefore PS is to SN as SN to SA. Q.E.D. COR. 1. PS2 is to SN2 as PS to SA. COR. 2. And because SA is given, SN- will be as PS. COR. 3. And the concourse of any tangent PM, with the right line SN. drawn from the focus per] endicular on the tangent, falls in the right line AN that touches the parabola in the principal vertex. PROPOSITION XIII. PROBLEM VIII. If a body moves in the perimeter of a parabola ; it is required to find the. law of the centripetal force tending to the focus of that figure. Retaining the construction of the preceding Lemma, let P be the body in the perimeter of the parabola ; and from the place Q,, into which it is next to succeed, draw QH parallel IS!. and Q,T perpendicular to SP, as also Qv parallel to the tan gent, and mating the diame ter PG in v, and the distance — 120 THE MATHEMATICAL PRINCIPLES [BOOK I. SP in x. Now. because of the similar triangles Pxv, SPM, and of the equal sides SP, SM of the one, the sides Px or Q,R and Pv of the other will be also equal. But (by the conic sections) the square of the ordinate Q,y is equal to the rectangle under the latus rectum and the segment Pv of the diameter ; that is (by Lem. XIII.), to the rectangle 4PS X Pv, or 4PS X Q,R ; and the points P and Q, coinciding, the ratio of Qv to Q,.r (by Cor. 2, Lem. VII.,) becomes a ratio of equality. And therefore Q,#2, in this case, becomes equal to the rectangle 4PS X Q,R. But (because of the similar triangles Q#T, SPN), Q^'2 is to QT2 as PS2 to SN2, that is (by Cor. 1, Lem. XIV.), as PS to SA ; that is, as 4PS X QR to 4SA x QR, and therefore (by Prop. IX. Lib. V., Elem.) QT* and 4SA X QR are SP2 SP2 X QT2 equal. Multiply these equals by ^-^-, and ^5 — -will become equal to SP2 X 4SA : and therefore (by Cor. 1 and 5, Prop. VL), the centripetal force is reciprocally as SP2 X 4S A ; that is, because 4SA is given, recipro cally in the duplicate ratio of the distance SP. Q.E.I. COR. 1. From the three last Propositions it follows, that if any body P goes from the place P with any velocity in the direction of any right line PR, and at the same time is urged by the action of a centripetal force that is reciprocally proportional to the square of the distance of the places from the centre, the body will move in one of the conic sections, having its fo cus in the centre of force ; and the contrary. For the focus, the point of contact, and the position of the tangent, being given, a conic section may be described, which at that point shall have a given curvature. But the curvature is given from the centripetal force and velocity of the body be ing given ; and two orbits, mutually touching one the other, cannot be de scribed by the same centripetal force and the same velocity. COR. 2. If the velocity with which the body goes from its place P is such, that in any infinitely small moment of time the lineola PR may be thereby describe I: and the centripetal force such as in the same time to move the same body through the space QR ; the body will move in one of QT2. the conic sections, whose principal latus rectum is the quantity Tjfr in its ultimate state, when thelineoke PR, QR are diminished in infinitum. In these Corollaries I consider the circle as an ellipsis ; and I except the case where the body descends to the centre in a right line. PROPOSITION XIV. THEOREM VI. Tf several bodies revolve about one common centre, and the centripetal force is reciprocally in tlie duplicate ratio of the distance of places from the centre ; I say, that the principal latera recta of tfieir orbits are in the duplicate ratio of the areas, which the bodies by radii drawn to the centre describe it\ the same time. SEC. HI. OF NATURAL PHILOSO1 HY. For (by Cor 2, Prop. XIII) the latus rectum QT*. L is equal to the quantity-^-in its ultimate state when the points P and Q, coincide. But the lineola QR in a given time is as the gen erating centripetal force ; that is (by supposi- QT2 tion), reciprocally as SP2. And therefore-^-^ is as Q.T2 X SP2 ; that is, the latus rectum L is in the duplicate ratio of the area QT X SP. Q.E.D. C?OR. Hence the whole area of the ellipsis, and the rectangle under the axes, which is proportional to it, is in the ratio compounded of the subdu- plicate ratio of the latus rectum, and the ratio of the periodic time. For the whole area is as the area QT X SP, described in a given time, mul tiplied by the periodic time. PROPOSITION XV. THEOREM VII. The same things being supposed, J say, that the periodic times in ellip ses are in the sesquiplicate ratio of their greater axes. For the lesser axis is a mean proportional between the greater axis and the latus rectum ; and, therefore, the rectangle under the axes is in the ratio compounded of the subduplicate ratio of the latus rectum and the sesquiplicate ratio of the greater axis. But this rectangle (by Cor. o. Prop. XIV) is in a ratio compounded of the subduplicate ratio of the latus rectum, and the ratio of the periodic time. Subduct from both sides the subduplicate ratio of the latus rectum, and there will remain the ses quiplicate ratio of the greater axis, equal to the ratio of the periodic time. Q.E.D. COR. Therefore the periodic times in ellipses are the same as in circles whose diameters are equal to the greater axes of the ellipses. PROPOSITION XVI. THEOREM VIII. The same things being supposed, and right lines being drawn to the bodies that shall touch the orbits, and perpendiculars being let fall on those tangents from the common focus ; I say, that the velocities oj the bodies are in a ratio compounded of the ratio of the perpendiculars inversely, and the, subduplicate ratio of the principal later a recta direct!]). From the focus S draw SY perpendicular to the tangent PR, and the velocity of the body P will be reciprocally in the subduplicate ratio of the SY2 quantity -y— . For that velocity is as the infinitely small arc PQ de- 122 THE MATHEMATICAL PRINCIPLES [BOOK I. scribed in a given moment of time, that is (by Lem. "VII), as the tangent PR ; that is (because of the proportionals PR to Q,T, and SP to SP X Q,T SY), as ~y — — ; or as SY reciprocally, and SP X Q,T directly ; but SP X QT is as the area described in the given time, that is (by Prop. XIV), in the subduplicate ratio of the latus rectum. Q.E.D. COR. 1. The principal latera recta are in a ratio compounded of the duplicate ratio of the perpendiculars and the duplicate ratio of the ve locities. COR. 2. The velocities of bodies, in their greatest and least distances from the common focus, are in the ratio compounded of the ratio of the distan ces inversely, and the subduplicate ratio of the principal latera recta di rectly. For those perpendiculars are now the distances. COR. 3. Arid therefore the velocity in a conic section, at its greatest or least distance from the focus, is to the velocity in a circle, at the same dis tance from the centre, in the subduplicate ratio of the principal latus rec tum to the double of that distance. COR. 4. The velocities of the bodies revolving in ellipses, at their mean distances from the common focus, are the same as those of bodies revolving in circles, at the same distances ; that is (by Cor. 6. Prop. IV), recipro cally in the subduplicate ratio of the distances. For the perpendiculars are now the lesser semi-axes, and these are as mean proportionals between the distances and the latera recta. Let this ratio inversely be compounded with the subduplicate ratio of the latera recta directly, and we shall have the subduplicate ratio of the distance inversely. COR. 5. In the same figure, or even in different figures, whose principal latera recta are equal, the velocity of a body is reciprocally as the perpen dicular let fall from the focus on the tangent. COR. 6. In a parabola, the velocity is reciprocally in the subduplicate ratio.of the distance of the body from the focus of the figure; it is more variable in the ellipsis, and less in the hyperbola, than according to this ratio. For (by Cor. 2, Lem. XIV) the perpendicular let fall from the focus on the tangent of a parabola is in the subduplicate ratio of the dis tance. In the hyperbola the perpendicular is less variable ; in the ellipsis more. COR. 7. In a parabola, the velocity of a body at any distance from the focus is to the velocity of a body revolving in a circle, at the same distance from the centre, in the subduplicate ratio of the number 2 to 1 ; in the ellipsis it is less, and in the hyperbola greater, than according to this ratio, For (by Cor. 2 of this Prop.) the velocitv at the vertex of a parabola is ir SEC. III.] OF NATURAL PHILOSOPHY. 123 this ratio, and (by Cor. 6 of this Prop, and Prop. IV) the same proportion holds in all distances. And hence, also, in a parabola, the velocity is everywhere equal to the velocity of a body revolving in a circle at half the distance ; in the ellipsis it is less, and in the hyperbola greater. COR. S. The velocity of a body revolving in any conic section is to the velocity of a body revolving in a circle, at the distance of half the princi pal latus rectum of the section, as that distance to the perpendicular let fall from the focus on the tangent of the section. This appears from Cor. 5. COR. 9. Wherefore since (by Cor. 6, Prop. IV), the velocity of a body revolving in this circle is to the velocity of another body revolving in any other circle reciprocally in the subduplicate ratio of the distances; there fore, ex czqiiO) the velocity of a body revolving *in a conic section will be to the velocity of a body revolving* in a circle at the same distance as a mean proportional between that common distance, and half the principal latus rectum of the section, to the perpendicular let fall from the common focus upon the tangent of the section. PROPOSITION XVII. PROBLEM IX. Supposing the centripetal force to be reciprocally proportional to the squares of the distances of places from the centre, and that the abso lute quantity of that force is known ; it is required to determine t/te line which a body will describe that is let go from a given place with a given velocity in the direction of a given right line. Let the centripetal force tending to the point S be such as will make the body p revolve in any given orbit pq ; and suppose the velocity of this body in the place p is known. Then from the place P suppose the body P to be let go with a given ve locity in the direction of the line PR ; but by virtue of a centripetal force to be immediately turned aside from that right line into the conic section PQ,. This, the right line PR will therefore touch in P. Suppose likewise that the right line pr touches the orbit pq in p ; and if from S you suppose perpendiculars let fall on those tangents, the principal latus rectum of the conic section (by Cor. 1, Prop. XVI) will be to the principal latus rectum of that orbit in a ratio compounded of the duplicate ratio of the perpendiculars, and the duplicate ratio of the velocities ; arid is therefore given. Let this latus rectum be L ; the focus S of the conic L24 THE MATHEMATICAL PRINCIPLES [BOOK I section is also given. Let the angle RPH be the complement of the angle RPS to two right ; and the line PH, in which the other focus II is placed, is given by position. Let fall SK perpendicular on PH, and erect the conjugate semi-axis BC ; this done, we shall have SP2— 2KPH + PH2 = SH2 = 4CH2 = 4BH2 — 4BC2 = SP + PH2— L X SiM SP2 + 2SPH + PH2 — L x SP + PH. Add on both sides 2KPH — SP2— PH2 + L X SP + PH, and we shall have L X SP + PH = 2SPH f 2KPH, or SP + PH to PH, as 2SP + 2KP to L. Whence PH is given both in length and position. That is, if the velocity of the body in P is such that the latus rectum L is less than 2SP + 2KP, PH will lie on the same side of the tangent PR writh the line SP ; and therefore the figure will be an ellipsis, which from the given foci S, H, and the principal axis SP + PH, is given also. But if the velocity of the body is so great, that the latus rectum L becomes equal to 2SP + 2KP, the length PH will be infinite ; and therefore, the figure will be a parabola, which has its axis SH parallel to the line PK, and is thence given. But if the body goes from its place P with a yet greater velocity, the length PH is to be taken on the other side the tangent ; and so the tangent pas sing between the foci, the figure will be an hyperbola having its principal axis equal to the difference of the lines SP and PH, and thence is given. Por if the body, in these cases, revolves in a conic section so found, it is demonstrated in Prop. XI, XII, and XIII, that the centripetal force will be reciprocally as the square of the distance of the body from the centre of force S ; and therefore we have rightly determined the line PQ,, which a body let go from a given place P with a given velocity, and in the di rection of the right line PR given by position, would describe with such a force. Q.E.F. COR. 1. Hence in every conic section, from the principal vertex D, the latus rectum L, and the focus S given, the other focus H is given, by taking DH to DS as the latus rectum to the difference between the latus rectum and 4US. For the proportion, SP + PH to PH as 2SP + 2KP to L, becomes, in the case of this Corollary, DS + DH to DH as 4DS to L, and by division DS to DH as 4DS — L to L. COR. 2. Whence if the velocity of a body in the principal vertex D ig given, the orbit may be readily found ; to wit, by taking its latus rectum to twice the distance DS, in the duplicate ratio of this given velocity to the velocity of a body revolving in a circle at the distance DS (by Cor. 3, Prop. XVI.), and then taking DH to DS as the latus rectum to the difference between the latus rectum and 4DS. COR. 3. Hence also if a body move in any conic section, and is forced out of its orbit by any impulse, you may discover the orbit in which it will afterwards pursue its Bourse. For bv compounding the proper motion oi SEC. IV.] OF NATURAL PHILOSOPHY. 125 the body with that motion, which the impulse alone would generate, you will have the motion with which the body will go off from a given place of impulse in the direction of a right line given in position. COR. 4. And if that body is continually disturbed by the action of some foreign force, we may nearly know its course, by collecting the changes which that force introduces in some points, and estimating the continual changes it will undergo in the intermediate places, from the analogy that appears in the progress of the series. SCHOLIUM. If a body P, by means of a centripetal force tending to any given point R, move in the perimeter of any given conic sec tion whose centre is C ; and the law of the centripetal force is required : draw CG parallel to the radius RP, and meet ing the tangent PG of the orbit in G ; and the force required (by Cor. 1, and CG3 Schol. Prop. X., and Cor. 3, Prop. VII.) will be as - SECTION IV. Of the finding of elliptic, parabolic, and hyperbolic orbits, from ttu. focus given. LEMMA XV. If from the two foci S, II, of any ellipsis or hyberbola, we draw to any third point V the right lines SV, H V, whereof one HV is equal to the principal axis of the figure, thai is, to the axis in which the foci are situated, the other, SV, is bisected in T by t/ie perpendicular TR let fall upon it ; that perpendicular TR will somewhere touch the conic section : and, vice versa, if it does touch it, HV will be equal to the principal axis of the figure. For, let the perpendicular TR cut the right line HV, produced, if need be, in R ; and join SR. Be cause TS, TV are equal, therefore the right lines SR, VR, as well as the angles TRS, TRV, will be also equal. Whence the point R will be in the conic section, and the perpen dicular TR will touch the same ; and the contrary. Q.E.D. 126 THE MATHEMATICAL PBINCIP, -ES [BOOK 1 PROPOSITION XVIII. PROBLEM X. From a focus and the principal axes given, to describe elliptic and hy perbolic trajectories, which shall pass through given points, and touch right lines given by position. Let S be the common focus of the figures ; AB A 33 the length of the principal axis of any trajectory ; r p T~* P a point through which the trajectory should \ /R pass ; and TR a right line which it should touch. / \ About the centre P, with the interval AB — SP, £\ S ~~yf if the orbit is an ellipsis, or AB {- SP, if the y> G ^ orbit is an hyperbola, describe the circle HG. On the tangent TR let fall the perpendicular ST, and produce the same to V, so that TV may be equal to ST; and about V as a centre with the interval AB describe the circle FH. In this manner, whether two points P, p, are given, or two tangents TR, tr, or a point P and a tangent TR, we are to describe two circles. Let H be their common intersection, and from the foci S, H, with the given axis describe the trajectory : I say, the thing is done. For (be cause PH -f- SP in the ellipsis, and PH — SP in the hyperbola, is equal to the axis) the described trajectory will pass through the point P, and (by the preceding Lemma) will touch the right line TR. And by the same argument it will either pass through the two points P, p, or touch the two right lines TR, tr. Q.E.F. PROPOSITION XIX. PROBLEM XI. About a given focus, to describe a parabolic trajectory, which shall pass through given points, and touch right lines given by position. Let S be the focus, P a point, and TR a tangent of the trajectory to be described. About P as a centre, with the interval PS, describe the circle FG. From the focus let fall ST perpendicular on the tangent, and produce the same to V, so as TV may be equal to ST. After the same manner another circle fg is to be de scribed, if another point p is given ; or another point v is to be found, if another tangent tr is given; then draw the right line IF, which shall touch the two circles YG,fg, if two points P, p are given ; or pass through the two points V, v, if two tangents TR, tr, are given : or touch the circle FG, and pass through the point V, if the point P and the tangent TR are given. On FI let fall the perpendicular SI, and bisect the same in K ; and with the axis SK and principal vertex K describe a parabola : I say the thing is done. For this parabola (because SK is equal to IK, and SP to FP) will pass through the point P ; and /KS SEC. IV.] OF NATURAL PHILOSOPHY. 127 (by Cor. 3, Lem. XIV) because ST is equal to TV. and STR a light an gle, it will touch the right line TR. Q.E.F. PROPOSITION XX. PROBLEM XII. About a given focus to describe any trajectory given in specie which shah pass through given points, and touch right lines given by position. CASE 1. About the focus S it is re- uired to describe a trajectory ABC, pass ing through two points B, C. Because the trajectory is given in specie, the ratio of the principal axis to the distance of the foci GAS H will be given. In that ratio take KB to BS, and LC to CS. About the centres B, C, with the intervals BK, CL, describe two circles ; and on the right line KL, that touches the same in K and L, let fall the perpendicu lar SG ; which cut in A and a, so that GA may be to AS, and Ga to aS, as KB to BS ; and with the axis A.«, and vertices A, a, describe a trajectory : I say the thing is done. For let H be the other focus of the described figure, and seeing GA is to AS as Ga to aS, then by division we shall have Ga — GA, or Aa to «S — AS, or SH in the same ratio; and therefore in the ratio which the principal axis of the figure to be described has to the distance of its foci ; and therefore the described figure is of the same species with the figure which was to be described. And since KB to BS, and LC to CS, are in the same ratio, this figure will pass through tht- points B, C, as is manifest from the conic sections. CASE 2. About the focus S it is required to describe a trajectory which shall somewhere touch two right lines TR, tr. From the focus on those tangents let fall the perpendiculars ST, St, which produce to V, v, so that TV, tv may be equal to TS, tS. Bisect Vv in O, and j erect the indefinite perpendicular OH, and cut I. the right line VS infinitely produced in K and V k, so that VK be to KS, and VA* to A~S, as the principal axis of the tra jectory to be described is to the distance of its foci. On the diameter K/J describe a circle cutting OH in H ; and with the foci S, H, and principal axis equal to VH, describe a trajectory : I say, the thing is done. For bisecting Kk in X, and joining HX, HS, HV, Hv, because VK is to KS as VA- to A*S ; and by composition, as VK -f- V/c to KS + kS ; and by division, as VA* — VK to kS — KS, that is, as 2VX to 2KX, and 2KX to 2SX, and therefore as VX to HX and HX to SX, the triangles VXH, HXS will be similar ; therefore VH will be to SH as VX to XH ; and therefore as VK to KS. Wherefore VH, the principal axis of the described trajectory, has the same ratio to SH, the distance of the foci, as 12S THE MATHEMATICAL PRINCIPLES [BOOK 1. K S the principal axis of the trajectory which was to be described has to the distance of its foci ; and is therefore of the same species. Arid seeing VH, vH are equal to the principal axis, and VS, vS are perpendicularly bisected by the right lines TR, tr, it is evident (by Lem. XV) that those right lines touch the described trajectory. Q,.E.F. CASE. 3. About the focus S it is required to describe a trajectory, which shall touch a right line TR in a given Point R. On the right line TR Jet fall the perpendicular ST, which produce to V, so that TV may be equal to ST ; join VR, and cut the right line VS indefinitely produced in K and k, so. that VK may be to SK, and V& to SAr, as the principal axis of the ellipsis to be described to the distance of its foci ; and on the diameter KA: describing a circle, cut the H right line VR produced in H ; then with the foci S, H, and principal axis equal to R VH, describe a trajectory : I say, the thing .---' is done. For VH is to SH as VK to SK, V" "1 and therefore as the principal axis of the trajectory which was to be de scribed to the distance of its foci (as appears from what we have demon strated in Case 2) ; and therefore the described trajectory is of the same species with that which was to be described ; but that the right line TR, by which the angle VRS is bisected, touches the trajectory in the point R, is certain from the properties of the conic sections. Q.E.F. CASE 4. About the focus S it is r required to describe a trajectory APB that shall touch a right line TR, and pass through any given point P without the tangent, and shall be similar to the figure apb, described with the principal axis ab, and foci s, h. On the tangent TR let fall the perpendicular ST, which / ..''''',.---"'" produce to V, so that TV may be equal to ST ; and making the an gles hsq, shq, equal to the angles VSP, SVP, about q as a centre, and with an interval which shall be to ab as SP to VS, describe a circle cut ting the figure apb in p : join sp, and draw SH such that it may be to sh as SP is to sp, and may make the angle PSH equal to the angle psh, and the angle VSH equal to the angle pyq. Then with the foci S, H, and B principal axis AB, equal to the distance VH, describe a conic section : I say, the thing is done ; for if sv is drawn so that it shall be to SEC. IV.] OF NATURAL PHILOSOPHY. 129 sp as sh is to sq, and shall make the angle vsp equal to the angle hsq, and the angle vsh equal to the angle psq, the triangles svh, spq, will be similar, and therefore vh will be to pq as sh is to sq ; that is (because of the simi lar triangles VSP, hsq), as VS is to SP? or as ab to pq. Wherefore vh and ab are equal. But, because of the similar triangles VSH, vsh, VH is to SH as vh to sh ; that is, the axis of the conic section now described is to the distance of its foci as the axis ab to the distance of the foci sh ; and therefore the figure now described is similar to the figure aph. But, because the triangle PSH is similar to the triangle psh, this figure passes through the point P ; and because VH is equal to its axis, and VS is per pendicularly bisected by the rght line TR, the said figure touches the right line TR. Q.E.F. LEMMA XVI. From three given points to draw to afonrth point that is not given three right lines whose differences shall be either given, or none at all. CASE 1. Let the given points be A, B, C, and Z the fourth point which we are to find ; because of the given difference of the lines AZ, BZ, the locus of the point Z will be an hyperbola whose foci are A and B, and whose princi pal axis is the given difference. Let that axis be MN. Taking PM to MA as MN is to AB, erect PR perpendicular to AB, and let fall ZR perpendicular to PR ; then from the nature of the hyperbola, ZR will be to AZ as MN is to AB. And by the like argument, the locus of the point Z will be another hyperbola, whose foci are A, C, and whose principal axis is the difference between AZ and CZ ; and QS a perpendicular on AC may be drawn, to which (QS) if from any point Z of this hyperbola a perpendicular ZS is let fall (this ZS), shall be to AZ as the difference between AZ and CZ is to AC. Wherefore the ratios of ZR and ZS to AZ are given, and consequently the ratio of ZR to ZS one to the other ; and therefore if the right lines RP, SQ, meet in T, and TZ and TA are drawn, the figure TRZS will be given in specie, and the right line TZ, in which the point Z is somewhere placed, will be given in position. There will be given also the right line TA, and the angle ATZ ; and because the ratios of AZ and TZ to ZS are given, their ratio to each other is given also ; and thence will be given likewise the triangle ATZ, whose vertex is the point Z. Q.E.I. CASE 2. If two of the three lines, for example AZ and BZ, are equal, draw the right line TZ so as to bisect the right line AB ; then find the triangle ATZ as above. Q.E.I. 130 THE MATHEMATICAL PRINCIPLES [BOOK I. CASE 3. If all the three are equal, the point Z will be placed in the centre of a circle that passes through the points A, B, C. Q.E.I. This problematic Lemma is likewise solved in Apollonius's Book oi Tactions restored by Vieta. PROPOSITION XXL PROBLEM XIII. About a given focus to describe a trajectory that shall pass through given points and touch right Hues given by position. Let the focus S, the point P, and the tangent TR be given, and suppose that the other focus H is to be found. On the tangent let fall the perpendicular ST, which produce to Y, so that TY may be equal to ST, and YH will be equal to the principal axis. Join SP, HP, and SP will be the difference between HP and the principal axis. After this manner, if more tangents TR are given, or more points P. we shall always determine as many lines YH, or PH, drawn from the said points Y or P, to the focus H, which either shall be equal to the axes, or differ from the axes by given lengths SP ; and therefore which shall either be equal among themselves, or shall have given differences ; from whence (by the preceding Lemma). that other focus H is given. But having the foci and the length of the axis (which is either YH, or, if the trajectory be an ellipsis, PH -f SP ; or PH — SP, if it be an hyperbola), the trajectory is given. Q.E.I. SCHOLIUM. When the trajectory is an hyperbola, I do not comprehend its conjugate hyperbola under the name of tins trajectory. For a body going on with a continued motion can never pass out of one hyperbola into its conjugate hyperbola. The case when three points are given is more readily solved thus. Let B, C, I), be the given points. Join BC, CD, and produce them to E, F, so as EB may be to EC as SB to SC ; and FC to FD as SC to SD. On EF drawn and pro duced let fall the perpendiculars SG, BH, and in GS produced indefinitely E take GA to AS, and Ga to aS, as HB is to BS ; then A will be the vertex, and Aa the principal axis of the tra jectory ; which, according as GA is greater than, equal to, or less than SEC. V.] OF NATURAL PHILOSOPHY. 131 AS. will be either an ellipsis, a parabola, or an hyperbola ; the point a in the first case falling on the same side of the line GP as the point A ; in the second, going oft* to an infinite distance ; in the third, falling on the other side of the line GP. For if on GF the perpendiculars CI, DK are let fall, TC will be to HB as EC to EB ; that is, as SO to SB ; and by permutation, 1C to SC as HB to SB, or as GA to SA. And, by the like argument, we may prove that KD is to SD in the same ratio. Where fore the points B, C, D lie in a conic section described about the focus S, in such manner that all the right lines drawn from the focus S to the several points of the section, and the perpendiculars let fall from the same points on the right line GF, are in that given ratio. That excellent geometer M. De la Hire has solved this Problem much after the same way, in his Conies, Prop. XXV., Lib. VIII. SECTION V. How the orbits are to be found when neither focus is given. LEMMA XVII. If from any point P of a given conic section, to the four produced sides AB, CD, AC, DB, of any trapezium ABDC inscribed in that section, as many right lines PQ, PR, PS, PT are drawn in given ang7ei, each line to each side ; the rectangle PQ, X PR of those on the opposite sides AB, CD, will be to the rectangle PS X PT of those on tie other two opposite sides AC, BD, in a given ratio. CASE 1. Let us suppose, first, that the lines drawn to one pair of opposite sides are parallel to either of I ^^ p ;T the other sides ; as PQ and PR to the side AC, and s| PS and PT to the side AB. And farther, that one pair of the opposite sides, as AC and BD, are parallel betwixt themselves; then the right line which bisects^ IQ I3 those parallel sides will be one of the diameters of the 1L conic section, and will likewise bisect RQ. Let O be the point in which RQ is bisected, and PO will be an ordinate to that diameter. Produce PO to K, so that OK may be equal to PO, and OK will be an ordinate on the other side of that diameter. Since, therefore, the points A, B; P and K are placed in the conic section, and PK cuts AB in a given angle, the rectangle PQK (by Prop. XVII., XIX., XXI. and XXI1L, Book III., of Apollonius's Conies) will be to the rectangle AQB in a given ratio. But QK and PR are equal, as being the differences of the equal lines OK, OP, and OQ, OR ; whence the rectangles PQK and PQ X PR are equal ; and therefore the rectangle PQ X PR is to the rectangle A^ B, that Is, to the rectangle PS X PT in a given ratio. Q.E.D 132 THE MATHEMATICAL PRINCIPLES [BOOK I CASE 2. Let us next suppose that the oppo site sides AC and BD of the trapezium are not parallel. Draw Be/ parallel to AC, and meeting as well the right line ST in /, as the conic section in d. Join Cd cutting PQ in r, and draw DM parallel to PQ, cutting Cd in M, and AB in N. Then (because of the similar triangles BTt, DBN), Et or PQ is to Tt as DN to NB. And ^^ Q N so Rr is to AQ or PS as DM to AN. Wherefore, by multiplying the antece- dents by the antecedents, and the consequents by the consequents, as the rectangle PQ X Rr is to the rectangle PS X Tt, so will the rectangle N i)M be to the rectangle ANB ; and (by Case 1) so is the rectangle PQ X Pr to the rectangle PS X Pt : and by division, so is the rectangle PQ X PR to the rectangle PS X PT. Q.E.D. CASE 3. Let us suppose, lastly, the four lines ?Q, PR, PS, PT, not to be parallel to the sides AC, AB, but any way inclined to them. In their place draw Pq, Pr, parallel to AC ; and Ps, Pt parallel to AB ; and because the angles of the triangles PQ</, PRr, PSs, PTt are given, the ra- tios of IQ to Pq, PR to Pr, PS to P*, PT to Pt will b? also given; and therefore the compound ed ratios Pk X PR to P? X Pr, and PS X PT to Ps X Pt are given. But from what we have demonstrated before, the ratio of Pq X Pi to Ps X Pt is given ; and therefore also the ratio of PQ X PR to PS X PT. Q.E.D. LEMMA XVIII. The s 'niL things supposed, if the rectangle PQ X PR of the lines drawn to the two opposite sides of the trapezium is to the rectangle PS X PT of those drawn to the other two sides in a given ratio, the point P, from whence those lines are drawn, will be placed in a conic section described about the trapezium. Conceive a conic section to be described pas sing through the points A, B, C, D, and any one of the infinite number of points P, as for example p ; I say, the point P will be always c1 placed in this section. If you deny the thing, join AP cutting this conic section somewhere else, if possible, than in P, as in b. Therefore if from those points p and b, in the given angles ^ B to the sides of the trapezium, we draw the right lines pq, pr, ps, pt, and bk, bn, bf, bd, we shall have, as bk X bn to bf X bd, SEC. V.] OF NATURAL PHILOSOPHY 133 so (by Lem. XVII) pq X pr to ps X pt ; and so (by supposition) PQ x PR to PS X PT. And because of the similar trapezia bkAf, PQAS, as bk to bf, so PQ to PS. Wherefore by dividing the terms of the preceding proportion by the correspondent terms of this, we shall have bn to bd as PR to PT. And therefore the equiangular trapezia ~Dnbd, DRPT, are similar, and consequently their diagonals D6, DP do coincide. Wherefore b falls in the intersection of the right lines AP, DP, and consequently coincides with the point P. And therefore the point P, wherever it is taken, falls to be in the assigned conic section. Q.E.D. COR. Hence if three right lines PQ, PR, PS, are drawn from a com mon point P, to as many other right lines given in position, AB, CD, AC, each to each, in as many angles respectively given, and the rectangle PQ X PR under any two of the lines drawn be to the square of the third PS in a given ratio ; the point P, from which the right lines are drawn, will be placed in a conic section that touches the lines AB; CD in A and C • and the contrary. For the position of the three right lines AB, CD, AC remaining the same, let the line BD approach to and coincide with the line AC ; then let the line PT come likewise to coincide with the line PS ; and the rectangle PS X PT will become PS2, and the right lines AB, CD, which before did cut the curve in the points A and B, C and D, can no (onger cut, but only touch, the curve in those coinciding points. SCHOLIUM. In this Lemma, the name of conic section is to be understood in a large sense, comprehending as well the rectilinear section through the vertex of the cone, as the circular one parallel to the base. For if the point p hap pens to be in a right line, by which the points A and D, or C and B are joined, the conic section will be changed into two right lines, one of which is that right line upon which the point p falls, and the other is a right line that joins the other two of *he four points. If the two opposite an gles of the trapezium taken together are equal c to two right angles, and if the four lines PQ, PR, PS, PT, are drawn to the sides thereof at right angles, or any other equal angles, and the rectangle PQ X PR under two of the lines drawn PQ and PR, is equal to the rectangle PS X PT under the other two PS and PT, the conic section will become a circle. And the same thing will happen if the four lines are drawn in any angles, and the rectangle PQ X PR, under one pair of the lines drawn, is to the rectangle PS X PT under the other pair as the rectangle under the sines of the angles S, T, in which the two last lines PS, PT are drawn to the rectangle under the sines of the angles Q, R, in which the first tw« 134 THE MATHEMATICAL PRINCIPLES [BOOK 1. PQ, PR are drawn. In all other cases the locus of the point P will be one of the three figures which pass commonly by the name of the conic sections. But in room of the trapezium A BCD, we may substitute a quadrilateral figure whose two opposite sides cross one another like diago nals. And one or two of the four points A, B, C, D may be supposed to be removed to an infinite distance, by which means the sides of the figure which converge to those points, will become parallel ; and in this case the conic section will pass through the other points, and will go the same way as the parallels in, infinitum. LEMMA XIX. To find a point P from which if four right lines PQ, PR, PS, PT an drawn to as many other right lines AB, CD, AC, BD, given by posi tion, each to each, at given angles, the rectangle PQ X PR, under any two of the lines' drawn, shall be to the rectangle PS X PT, under the other tivo. in a given ratio. Suppose the lines AB, CD, to which the two right lines PQ, PR, containing one of the rect angles, are drawn to meet two other lines, given by position, in the points A, B, C, D. From one of those, as A, draw any right line AH, in which you would find the point P. Let this cut the opposite lines BD, CD, in H and I ; and, because all the angles of the figure are given, the ratio of PQ to PA, and PA to PS, and therefore of PQ to PS, will be also given. Subducting this ratio from the given ratio oi PQ X PR to PS X PT, the ratio of PR to PT will be given ; and ad ding the given ratios of PI to PR, and PT to PH, the ratio of PI to PH. and therefore the point P will be given. Q.E.I. COR. 1. Hence also a tangent may be drawn to any point D of the locus of all the points P. For the chord PD, where the points P and D meet, that is, where AH is drawn through the point D, becomes a tangent. In which case the ultimate ratio of the evanescent lines IP and PH will be found as above. Therefore draw CF parallel to AD, meeting BD in F, and cut it in E in the same ultimate ratio, then DE will be the tan gent ; because CF and the evanescent IH are parallel, and similarly cut in E and P. COR. 2. Hence also the locus of all the points P may be determined. Through any of the points A, B, C, D, as A, draw AE touching the locus, and through any other point B parallel to the tangent, draw BF meeting the locus in F ; and find the point F by this Lemma. Bisect BF in G, and, drawing the indefinite line AG, this will be the position of the dia meter to which BG and FG are ordinates. Let this AG meet the locus SEC. V.J OF NATURAL PHILOSOPHY. in H, and AH will be its diameter or latus trans- versum. to which the latus rectum will be as BG2 to AG X GH. If AG nowhere meets the locus, the line AH being infinite, the locus will be a par abola ; and its latus rectum corresponding to the diameter AG will be -.-7^ AC* But if it does meet it anywhere, the locus will be an hyperbola, when the points A and H are placed on the same side the point G ; and an ellipsis, if the point G falls between the points A and H ; unless, perhaps, the angle AGB is a right angle, and at the same time BG2 equal to the rectangle AGH, in which case the locus will be a circle. And so we have given in this Corollary a solution of that famous Prob lem of the ancients concerning four lines, begun by Euclid, and carried on by Apollonius ; and this not an analytical calculus, but a geometrical com position, such as the ancients required. LEMMA XX. If the two opposite angular points A and P of any parallelogram ASPQ touch any conic section in the points A and P ; and the sides AQ, AS of one of those angles, indefinitely produced, meet the same conic section in B and C ; and from the points of concourse, B and C to any fifth point D of the conic section, two right lines BD, CD are drawn meet- ing tlie two other sides PS, PQ of the parallelogram, indefinitely pro duced in T and R ; the parts PR and PT, cut off from the sides, will always be one to the other in a given ratio. And vice versa, if those parts cut off' are one to the other in a given ratio, the locus of the point D will be a conic section passing through the four points A, B, C, F CASE 1. Join BP, CP, and from the point D draw the two right lines DG, DE, of which the first DG shall be parallel to AB, and meet PB, PQ, CA in H, I, G ; and the other DE shall be parallel to AC, and meet PC, PS, AB, in F, K, E ; and (by Lem. XVII) the rectangle DE X DF will be to the rect angle DG X DH in a given ratio. But PQ is to DE (or IQ) as PB to HB, and con sequently as PT to DH ; and by permutation PQ, is to PT as DE to DH. Likewise PR is to DF as RC to DC, and therefore as (IG or) PS to DG ; and by permutation PR is to PS as DF to DG ; and, by com pounding those ratios, the rectangle PQ X PR will be to the rectangle PS X PT as the rectangle DE X DF is to the rectangle DG X DH. and consequently in "a given ratio. But PQ and PS are given, and there fore the ratio of PR to PT is given. Q.E.D. 136 THE MATHEMATICAL PRINCIPLES CASE 2. But if PR and PT are supposed to be in a given ratio one to the other, then by going back again, by a like reasoning, it will follow that the rectangle DE X DF is to the rectangle DG X DH in a given rati) ; and so the point D (by Lem. XVIII) will lie in a conic section pass ing through the points A., B, C, P, as its locus. Q.E.I). COR. 1. Hence if we draw BC cutting PQ in r and in PT take Pt to Pr in the same ratio which PT has to PR ; then Et will touch the conic section in the point B. For suppose the point D to coalesce with the point B, so that the chord BD vanishing, BT shall become a tangent, and CD and BT will coincide with CB and Bt. COR. 2. And, vice versa, if Bt is a tangent, and the lines BD, CD meet in any point D of a conic section, PR will be to PT as Pr to Pt. And, on the contrary, if PR is to PT as Pr to Pt, then BD and CD will meet in some point D of a conic section. COR. 3. One conic section cannot cut another conic section in more than four points. For, if it is possible, let two conic sections pass through the h've points A, B, C, P, O ; and let the right line BD cut them in the points D, d, and the right line Cd cut the right line PQ, in q. Therefore PR is to PT as Pq to PT : whence PR and Pq are equal one to the other, against the supposition. LEMMA XXI. If two moveable and indefinite right lines BM, CM drawn through given points B, C, as poles, do by their point of concourse M describe a third right line MN given by position ; and other two indefinite right lines BD,CD are drawn, making with the former two at those given points B, C, given angles, MBD, MCD : I say, that those two right lines BD, CD will by their point of concourse D describe a conic section passing through the points B, C. And, vice versa, if the right lints BD, CD do by their point of concourse D describe a conic section passing through the given points B, C, A, and the angle DBM is always equal to the giren angle ABC, as well as the angle DCM always equal to the given angle ACB, the point M will lie in a right line given by position, as its locus. For in the right line MN let a point N be given, and when the moveable point M falls on the immoveable point N. let the moveable point D fall on an immo vable point P. Join ON, BN, CP, BP, and from the point P draw the right lines PT, PR meeting BD, CD in T and R, C and making the angle BPT c jual to the given angle BNM, and the angle CPR SEC. V.J OF NATURAL PHILOSOPHY. 137 equal to the given angle CNM. Wherefore since (by supposition) the an gles MBD, NBP are equal, as also the angles MOD, NCP, take away the angles NBD and NOD that are common, and there will remain the angles NBM and PBT, NCM and PCR equal; and therefore the triangles NBM, PBT are similar, as also the triangles NCM, PCR. Wherefore PT is to NM as PB to NB ; and PR to NM as PC to NC. But the points, B, C, N, P are immovable: wheiefore PT and PR have a given ratio to NM, and consequently a given ratio between themselves; and therefore, (by Lemma XX) the point D wherein the moveable right lines BT and CR perpetually concur, will be placed in a conic section passing through the points B. C, P. Q.E.D. And, vice versa, if the moveable point D lies in a conic section passing through the given points B, C, A ; and the angle DBM is always equal to the given an gle ABC, and the angle DCM always equal to the given angle ACB, and when the point D falls successively on any two immovable points p, P, of the conic section, the moveable point M falls suc cessively on two immovable points /?, N. Through these points ??, N, draw the right line nN : this line nN will be the perpetual locus of that moveable point M. For, if possible, let the point M be placed in any curve line. Therefore the point D will be placed in a conic section passing through the five points B, C, A, p, P, when the point M is perpetually placed in a curve line. But from what was de monstrated before, the point D will be also placed in a conic section pass ing through the same five points B, C, A, p, P, when the point M is per petually placed in a right line. Wherefore the two conic sections will both pass through the same five points, against Corol. 3, Lem. XX. It is therefore absurd to suppose that the point M is placed in a curve line. QE.D. PROPOSITION XXII. PROBLEM XIV. To describe a trajectory that shall pass through Jive given points. Let the five given points be A, B, C, P, D. c From any one of them, as A, to any other sv two as B, C, which may be called the poles, draw the right lines AB, AC, and parallel to those the lines TPS, PRO, through the fourth point P. Then from the two poles B, C, draw through the fifth point D two indefinite lines BDT, CRD, meeting with the last drawn lines TPS, PRQ (the 138 THE MATHEMATICAL PRINCIPLES IBOOK L former with the former, and the latter with the latter) in T and R. Then drawing the right line tr parallel to TR, cutting off from the right lines PT, PR, any segments Pt, Pr, proportional to PT, PR ; and if through their extremities, t, r, and the poles B, C, the right lines lit, Cr are drawn, meeting in d, that point d will be placed in the trajectory required. For (by Lena. XX) that point d is placed in a conic section passing through the four points A, B, C, P ; and the lines R/', TV vanishing, the point d comes to coincide with the point D. Wherefore the conic section passes through the five points A, B, C, P, D. Q.E.D. The same otherwise. Of the given points join any three, as A, B, C ; and about two of them 15, C, as poles, making the angles ABC, ACB of a given magnitude to revolve, apply the legs BA, CA, first to the point D, then to the point P, and mark the points M, N, in which the other legs BL, CL intersect each other in both cases. C' Draw the indefinite right line MN, and let those moveable angles revolve about their poles B, C, in such manner that the intersection, which is now supposed to be ???, of the legs BL, CL; or BM7 CM, may always fall in that indefinite right line MN ; and the intersection, which is now supposed to be d, of the legs BA ^A, or BD; CD, will describe the trajectory required, PADc/B. For (by Lem. XXI) the point d will be placed in a conic section passing- through the points B, C ; and when the point m comes to coincide with the points L, M, N, the point d will (by construction) come to coin cide with the points A, D, P. Wherefore a conic section will be described that shall pass through the five points A, B. C, P, D. Q,.E.F. COR. 1. Hence a right line may be readily drawn which shall be a tan gent to the trajectory in any given point B. Let the point d come to co incide with the point B, arid the right line Bt/ Avill become the tangent required. COR. 2. Hence also may be found the centres, diameters, and latera recta of the trajectories, as in Cor. 2, Lem. XIX. SCHOLIUM. The former of these constructions will be- c come something more simple by joining , and in that line, produced, if need be, aking Bp to BP as PR is to PT ; and t rough p draw the indefinite right inc j0e parallel to S PT, and in that line pe taking always pe equal to Pi , and draw the right lines Be, Cr SEC. Y.J OF NATURAL PHILOSOPHY. 139 to meet in d. For since Pr to Pt, PR to PT, pB to PB, pe to Pt, are all in the same ratio, pe and Pr will be always equal. After this manner the points of the trajectory are most readily found, unless you would rather describe the curve mechanically, as in the second construction. PROPOSITION XXIII. PROBLEM XV. To describe a trajectory that shall pass through four given points, and touch a right line given by position. CASE 1. Suppose that HB is the given tangent, B the point of contact, and C, 1., P, the three other given points. Jo n BC. and draw IS paral lel to BH, and PQ parallel to BC ; complete the parallelogram BSPQ. Draw BD cutting SP in T, and CD cutting PQ, in R. Lastly, draw any line tr parallel to TR, cutting off from PQ, PS, the segments Pr, Pt proportional to PR, PT respectively ; and draw Cr, Bt their point of concourse d will (by Lem. XX) always fall on the trajectory to be described. The same otherwise. 1 et tl e angle CBH of a given magnitude re volve about the pole B; as also the rectilinear ra- d:us 1C, both ways produced, about the pole C. Mark the points M, N, on which the leg BC of the angle cuts that radius when BH; the other leg thereof, meets the same radius in the points P and D. Then drawing the indefinite line MN, let that radius CP or CD and the leg BC of the angle perpetually meet in this Ikie; and the point of concourse of the other leg BH with the radius will delineate the trajectory required. For if in the constructions of the preceding Problem the point A comes to a coincidence with the point B, the lines CA and CB will coincide, and the line AB, in its last situation, will become the tangent BH ; and there fore the constructions there set down will become the same with the con structions here described. Wherefore the concourse of the leg BH with the radius will describe a conic section passing through the points C, D, P, and touching the line BH in the point B. Q.E.F. CASE 2. Suppose the four points B, C, D, P, given, being situated with- ont the tangent HI. Join each two by the lines BD, CP meeting in G, and cutting the tangent in H and I. Cut the tangent in A in such mannr: 140 THE MATHEMATICAL PRINCIPLES [BOOK I X IT that HA may be to IA as the rectangle un der a mean proportional between CG and GP, and a mean proportional between BH and HD is to a rectangle under a mean pro portional between GD and GB, and a mean proportional betweeen PI and 1C, and A will be the point of contact. For if HX, a par allel to the right line PI, cuts the trajectory in any points X and Y, the point A (by the properties of the conic sections) will come to be so placed, that HA2 will become to AP in a ratio that is compounded out of the ratio of the rec tangle XHY to the rectangle BHD, or of the rectangle CGP to the rec tangle DGB; and the ratio of the rectangle BHD to the rectangle PIC. But after the point of contac.t A is found, the trajectory will be described as in the first Case. Q.E.F. But the point A may be taken either between or without the points H and I, upon which account a twofold trajectory may be described. PROPOSITION XXIV. PROBLEM XVI. To describe a trajectory that shall pass through three given points, and touch two right lines given by position. Suppose HI, KL to be the given tangents and B, C, D, the given points. Through any two of those points, as B, D, draw the indefi nite right line BD meeting the tangents in the points H, K. Then likewise through any other two of these points, as C, D, draw the indefinite right line CD meeting the tan gents in the points I, L. Cut the lines drawn in R and S, so that HR may be to KR as the mean proportional between BH and HD is to the mean proportional between BK and KD ; and IS to LS as the mean pioportional between CI and ID is to the mean proportional between CL and LD. But you may cut, at pleasure, either within or between the points K and H, I and L, or without them ; then draw RS cutting the tangents in A and P, and A and P will be the points of contact. For if A and P are supposed to be the points of contact, situated anywhere else in the tangents, and through any of the points H, I, K, L, as I, situated in either tangent HI, a right line IY is drawn parallel to the other tangent KL, and meeting the curve in X and Y, and in that right line there be taken IZ equal to a mean pro portional between IX and IY, the rectangle XIY or IZ2, will (by the pro perties of the conic sections) be to LP2 as the rectangle CID is to the rect angle CLD, that is (by the construction), as SI is to SL2; and therefore SEC. V.] OF NATUKAL PHILOSOPHY. 141 IZ is to LP as SI to SL. Wherefore the points S, P, Z. are in one right line. Moreover, since the tangents meet in G, the rectangle XI Y or IZ2 will (by the properties of the conic sections) be to IA2 as GP2 is to GA2, and consequently IZ will be to I A as GP to GA. Wherefore the points P, Z, A, lie in one right line, and therefore the points S, P, and A are in one right line. And the same argument will prove that the points R, P, and A are in one right line. Wherefore the points of contact A and P lie in the right line RS. But after these points are found, the trajectory may be described, as in the first Case of the preceding Problem. Q,.E.F. In this Proposition, and Case 2 of the foregoing, the constructions are the same, whether the right line XY cut the trajectory in X and Y, or not ; neither do they depend upon that section. But the constructions being demonstrated where that right line does cut the trajectory, the con structions where it does not are also known ; and therefore, for brevity's sake, I omit any farther demonstration of them. LEMMA XXII. To transform figures into other figures of the same kind. Suppose that any figure HGI is to be transformed. Draw, at pleasure, two par allel lines AO, BL, cutting any third line AB, given by position, in A and B, and from any point G of the figure, draw out any right line GD, parallel to OA, till it meet the right line AB. Then from any given point 0 in the line OA, draw to the point D the right line OD, meeting BL in d ; and from the point of concourse raise the right line dg containing any given angle with the right line BL, and having such ratio to Qd as DG has to OD ; and g will be the point in the new figure hgi, corresponding to the point G. And in like manner the several points of the first figure will give as many correspondent points of the new figure. If we therefore conceive the point G to be carried along by a con tinual motion through all the points of the first figure, the point g will be likewise carried along by a continual motion through all the points of the new figure, and describe the same. For distinction's sake, let us call DG the first ordinate, dg the new ordinate, AD the first abscissa, ad the new abscissa ; O the pole. OD the abscinding radius, OA the first ordinate radius, and Oa (by which the parallelogram OABa is completed) the new ordinate radius. I say, then, that if the point G is placed in a right line given by posi tion, the point g will be also placed in a right line given by position. If the point G is placed in a conic section, the point g will be likewise placed J42 THE MATHEMATICAL PRINCIPLES [BOOK 1. in a conic section. And here I understand the circle as one of the conic sections. But farther, if the point G is placed in a line of the third ana lytical order, the point g will also be placed in a line of the third order, and so on in curve lines of higher orders. The two lines in which the points G, g, are placed, will be always of the same analytical order. For as ad is to OA, so are Od to OD, dg to DG, and AB to AD ; and there- OA X AB OA X dg fore AD is equal to , , and DG equal to 7— — . Now if the ad ad point G is placed in a right line, and therefore, in any equation by which the relation between the abscissa AD and the ordinate GD is expressed, those indetermined lines AD and DG rise no higher than to one dimen- v v xu- ,. OA X AB . OA X dg sion, by writing this equation— . m place of AD, and — — -. — - in place of DG, a new equation will be produced, in which the new ab scissa ad and new ordinate dg rise only to one dimension ; and which therefore must denote a right line. But if AD and DG (or either of them) had risen to two dimensions in the first equation, ad and dg would likewise have risen to tAvo dimensions in the second equation. And so on in three or more dimensions. The indetermined lines, ad} dg in the second equation, and AD, DG, in the first, will always rise to the same number of dimensions ; and therefore the lines in which the points G, g, are placed are of the same analytical order. I say farther, that if any right line touches the curve line in the first figure, the same right line transferred the same way with the curve into the new figure will touch that curve line in the new figure, and vice versa. For if any two points of the curve in the first figure are supposed to ap proach one the other till they come to coincide, the same points transferred will approach one the other till they come to coincide in the new figure ; and therefore the right lines with which those points are joined will be come together tangents of the curves in both figures. I might have given demonstrations of these assertions in a more geometrical form ; but I study to be brief. Wherefore if one rectilinear figure is to be transformed into another, we need only transfer the intersections of the right lines of which the first figure consists, and through the transferred intersections to draw right lines in the new figure. But if a curvilinear figure is to be transformed, we must transfer the points, the tangents, and other right lines, by means of which the curve line is denned. This Lemma is of use in the solution of the more difficult Problems ; for thereby we maj' transform the proposed figures, if they are intricate, into others that are more simple. Thus any right lines converging to a point are transformed into parallels, by taking for the first ordinate radius any right line that passes through the point of concourse of the converging lines, and that because their point of con- SEC. V.] OF NATURAL PHILOSOPHY. 143 course is by this means made to go off in infinitum ; and parallel lines are such as tend to a point infinitely remote. And after the problem is solved in the new figure, if by the inverse operations we transform the new into the first figure, we shall have the solution required. This Lemma is also of use in the solution of solid problems. For as often as two conic sections occur, by the intersection of which a problem may be solved, any one of them may be transformed, if it is an hyperbola or a parabola, into an ellipsis, and then this ellipsis may be easily changed into a circle. So also a right line and a conic section, in the construc tion of plane problems, may be transformed into a right line and a circle PROPOSITION XXV. PROBLEM XVII. To describe a trajectory that shall pass through two given points, and touch three right lines given by position. Through the concourse of any two of the tangents one with the other, and the concourse of the third tangent with the right line which passes through the two given points, draw an indefinite right line ; and, taking this line for the first ordinate radius, transform the figure by the preceding Lemma into a new figure. In this figure those two tangents will become parallel to each other, and the third tangent will be parallel to the right line that passes through the two given points. Suppose hi, kl to be those two parallel tangents, ik the third tangent, and hi a right line parallel thereto, passing through those points a, b, through which the conic section ought to pass in this new figure; and completing the paral- lelogra n fiikl, let the right lines hi, ik, kl be BO cut in c, d, e, that he may be to the square root of the rectangle ahb, ic, to id, and ke to kd. as the sum of the right lines hi and kl is to the sum of the three lines, the first whereof ' is the right line ik, and the other two are the square roots of the rectangles ahb and alb ; and c, d, e, will be the points of contact. For by the properties of the conic sections, he2 to the rectan gle ahb, and ic2 to id2, and ke2 to kd2, and el2 to the rectangle alb, are all in the same ratio ; and therefore he to the square root of ahb, ic to id, ke to kdj and el to the square root of alb, are in the subduplicate of that ratio ; and by composition, in the given ratio of the sum of all the ante cedents hi + kly to the sum of all the consequents ^/ahb -\- ik : *Jalb, Wherefore from that given ratio we have the points of contact c, d, e, in the new figure. By the inverted operations of the last Lemma, let those points be transferred into the first figure, and the trajectory will be there described by Prob. XIV. Q.E.F. But according as the points a, b, fall between the points //, /, or without taem, the points c, d, e, must be taken 144 THE MATHEMATICAL PRINCIPLES BOOK I.J Cither between the points, h, i, k, /, or without them. If one of the points a, b, falls between the points h, i, and the other xvithout the points h, I, the Problem is impossible. PROPOSITION XXVI. PROBLEM XVIII. To describe a trajectory that shall pass through a given point, and touch four right lines given by position. From the common intersections, of any two of the tangents to the common intersection of the other two, draw an indefinite right line ; and taking this line for the first ordinate radius; / 'xs o transform the figure (by Lem. XXII) into a new figure, and the two pairs of tangents, each of which before concurred in the first ordinate ra- dius, will now become parallel. Let hi and kl, Al l\ ik and hi, be those pairs of parallels completing the parallelogram hikl. And let p be the point in this new figure corresponding to the given point in the first figure. Through O the centre of the figure draw pq.: and O? being equal to Op, q will be the other point through which the conic sec tion must pass in this new figure. Let this point be transferred, by the inverse operation of Lem. XXII into the first figure, and there we shall have the two points through which the trajectory is to be described. But through those points that trajectory may be described by Prop. XVII. LEMMA XXIII. If two right lines, as AC, BD given by position, and terminating in given points A, B, are in a given ratio one to the other, and the right line CD, by which the, indetermined points C, D are joined is cut in K in a given ratio ; I say, that the point K will be placed in a right line given by position. For let the right lines AC, BD meet in E, and in BE take BG to AE as BD is to AC, and let FD be always equal to the given line EG ; and, by construction, EC will be to GD, that is, to EF, as AC to BD, and therefore in a given ratio ; and therefore the %•-'''' ,.---'' I \ triangle EFC will be given in kind. Let E K cT^"^ CF be cut in L so as CL may be to CF in the ratio of CK to CD ; and because that is a given ratio, the triangle EFL will be given in kind, and therefore the point L will be placed in the right line EL given by position. Join LK, and the triangles CLK, CFD will be similar ; and because FD is a given line, and LK is to FD in a given ratio, LK will be also given SEC. V.] OF NATURAL PHILOSOPHY. 145 To this let EH be taken equal, and ELKH will be always a parallelogram. And therefore the point K is always placed in the side HK (given by po tiition) of that parallelogram. Q.E.D. COR. Because the figure EFLC is given in kind, the three right lines EF, EL, and EC, that is, GD, HK, and EC, will have given ratios to each other. LEMMA XXIV. If three right lines, two whereof are parallel, and given by position, touch any conic section ; I say, that the semi-diameter of the section wkiJt is parallel to those two is a mean proportional between the segments of those two that are intercepted between the points of contact and the. third tangent. Let AF, GB be the two parallels touch ing the conic section ADB in A and B ; EF the third right line touching the conic section in I, and meeting the two former tangents in F and G, and let CD be the semi-diameter of the figure parallel to those tangents ; I say. that AF, CD, BG are continually proportional. For if the conjugate diameters AB, DM G Q meet the tangent FG in E and H, and cut one the other in C; and the parallelogram IKCL be completed ; from the nature of the conic sections, EC will be to CA as CA to CL ; and so by division, EC — CA to CA -— CL, orEAto AL; and by composition, EA to EA + AL or EL, as EC to EC + CA or EB ; and therefore (because of the similitude of the triangles EAF, ELI, ECH, EBG) AF is to LI as CH to BG. Likewise, from tli? nature of the conic sections, LI (or CK) is to CD as CD to CH ; and therefore (ex aquo pertnrhatfy AF is to CD as CD to BG. Q.E.D. COR. 1. Hence if two tangents FG, PQ, meet two parallel tangents AF, BG in F and G, P and Q,, and cut one the other in O; AF (ex cequo per- tnrbot, ) will be to BQ as AP to BG, and by division, as FP to GQ, and therefore as FO to OG. COR. 2. Whence also the two right lines PG, FQ, drawn through the points P and G, F and Q, will meet in the right line ACB passing through the centre of the figure and the points of contact A, B. LEMMA XXV. If four sides of a parallelogram indefinitely produced touch any conic section, and are cut by a fifth tangent ; I say, that, taking those seg ments of any two conterminous sides that terminate in opposite angles 10 146 THE MATHEMATICAL PRINCIPLES [BooK 1. of the parallelogram, either segment is to the side from which it is cut off as that part of the other conterminous side which is intercepted between the point of contact and the third side is to Uie other segment, Let the four sides ML, IK, KL, MI, of the parallelogram ML JK touch the conic section in A, B, C, I) ; and let the fifth tangent FQ cut those sides in F, Q, H, and E : and taking the segments ME, KQ of the sides Ml, KJ, or the segments KH, MF of the sides KL, ML, 1 s/.y, that ME is to MI as BK to KQ; and KH to KL as AM to MF. For, by Cor. 1 of the preceding Lemma, ME is to El as (AM or) BK to BQ ; and, by composition, ME is to MI as BK to KQ. Q.E.D. Also KH is to HL as (BK or) AM to AF ; and by division, KH to KL as AM to MF. Q.E.D. COR. 1. Hence if a parallelogram IKLM described about a given conic section is given, the rectangle KQ X ME, as also the rectangle KH X ME equal thereto, will be given. For, by reason of the similar triangles KQH MFE, those rectangles are equal. COR. 2. And if a sixth tangent eq is drawn meeting the tangents Kl. MI in q and e, the rectangle KQ X ME will be equal to the rectangle K</ X Me, and KQ will be to Me as Kq to ME, and by division ns Q? to Ee. COR. 3. Hence, also, if E<?, eQ, are joined and bisected, and a right line is drawn through the points of bisection, this right line will pass through the centre of the conic section. For since Q</ is to Ee as KQ to Me, the same right line will pass through the middle of all the lines Eq, eQ, MK (by Lem. XXIII), and the middle point of the right line MK is the centre of the section. PROPOSITION XXVII. PROBLEM XIX. To describe a trajectory that may touch jive right lines given by position. Supposing ABG; BCF, GCD, FDE, EA to be the tangents given by position. Bisect in M and N, AF, BE, the diagonals of the quadri lateral tained figure under ABFE con- any four of them ; and (by Cor. 3, Lem. XXV) the right line MN draAvn through the points (,f SEC. V.] OF NATURAL PHILOSOPHY. 147 bisection will pass through the centre of the trajectory. Again, bisect in P and Q, the diagonals (if I may so call them) Bl), GF of the quadrila teral figure EC OF contained under any other four tangents, and the right line PQ, drawn through the points of bisection will pass through the cen tre of the trajectory ; and therefore the centre will be given in the con course of the bisecting lines. Suppose it to be O. Parallel to any tan gent BG draw KL at such distance that the centre O may be placed in the middle between the parallels; this KL will touch the trajectory to be de scribed. Let this cut any other two tangents GCD, FJ)E, in L and K. Through the points G and K, F and L, where the tangents not parallel, CL, FK meet the parallel tangents CF, KL, draw GK, FL meeting in K ; and the right line OR drawn and produced, will cut the parallel tan gents GF, KL, in the points of contact. This appears from Gor. 2, Lem. XXIV. And by the same method the other points of contact may be found, and then the trajectory may be described by Prob. XIV. Q.E.F. SCPIOLTUM. Under the preceding Propositions are comprehended those Problems wherein either the centres or asymptotes of the trajectories are given. For when points and tangents and the centre are given, as many other points and as many other tangents are given at an equal distance on the other side of the centre. And an asymptote is to be considered as a tangent, ami its infinitely remote extremity (if we may say so) is a point of contact. Conceive the point of contact of any tangent removed in infinitum, and the tangent will degenerate into an asymptote, and the constructions of the preceding Problems will be changed into the constructions of those Problems wherein the asymptote is given. After the trajectory is described, we may find its axes and foci in this manmr. In the construction and figure of Lem. XXI, let those , legs BP, CP, of the moveable angles PEN, ^ PCN, by the concourse of which the trajec- \ tory was described, be made parallel one to the other : and retaining that position, let them revolve about their poles I', C, in that figure. In the mean while let the other legs GN, BN, of those angles, by their concourse K or k, describe the circle BKGC. Let O be the centre of this circle; and from this centre upon the ruler MN, wherein those legs CN, BN did concur while the trajectory was described, let fall the perpendicular OH meeting the circle in K and L. And when those other legs CK, BK meet in the point K that is nearest to the ruler, the first legs CP, BP will be parallel to the greater axis, and perpendicular on the lesser ; and the con- 148 THE MATHEMATICAL PRINCIPLES [Book I trary will hajpen if those legs meet in the remotest point L. 'Whence ii the centre of the trajectory is given, the axes will be given ; and those be- ing given, the foci will be readily found. But the squares of the axes are one to the other as KH to LH, and thence it is easy to describe a trajectory given in kind through f mr given points. For if two of the given points are made the poles C, 13, the third will give the moveable angles PCK, PBK ; but those being given, the circle BGKC may be described. Then, because the trajectory is given in kind, the ratio of OH to OK, and and therefore OH itself, will be given. About ' the centre O, with the interval OH, describe another circle, and the right line that touches this circle, and passes through the concourse of the legs CK, BK, when the first legs CP; BP meet in the fourth given point, will be the ruler MN, by means of which the trajectory may be described Whence also on the other hand a trapezium given in kind (excepting a few cases that are impossible) may be inscribed in a given conic section. There are also other Lemmas, by the help of which trajectories given in kind may be described through given points, and touching given lines. Of such a sort is this, that if a right line is drawn through any point given by position, that may cut a given conic section in two points, and the distance of the intersections is bisected, the point of bisection will to'ich ano her conic section of the same kind with the former, arid havin^ its axes parallel to the axes of the former. But I hasten to things of greater use. LEMMA XXVI. To place 1ht' lit rev angles of a triangle, given both in kind and magni tude, in, respect of as many rigid lines given by position, -provided th°\] are not all parallel among themselves, in such manner tfia' tjic spiral angles may touch the several lines. Three indefinite right lines AB, AC, BC, are given by position, and it is required so to place the triangle DEF that its angle 1) may touch the line AB, its angle E the line AC, and its angle F the line BC. Upon DE, DF, and EF, describe three segments of circles DRE, DGF. EMF, capable of angles equal to the Rubles BAG, ABC, ACB respectively. But those segments are to be de scribed t« wards such sides of the lines DE, DF; EF; that the letters 3 EC. V.I OF NATURAL PHILOSOPHY. 1411 DRED may turn round about in the same order with the letters I1ACB : the letters DGFD in the same order with the letters ABCA ; and the letters EMFE in the same order with the letters ACBA ; then, completing th 'se segmerts into entire circles let the two former circles cut one the other in G, and suppose P and Q to be their centres. Then joining GP, PQ, take Ga to AB as GP is to PQ ; and about the centre G, with the interval Ga, describe a circle that may cut the first circle DGE in a. Join aD cutting the second circle DFG in b, as well as aE cutting the third circle EMF in c. Complete the figure ABCdef similar and equal to the figure a&cDEF : I say, the thing is done. For drawing Fc meeting «D in n, and joining aG; bG, QG, QD. PD, by construction the angle EaD is equal to the angle CAB, and the angle acF equal to the angle ACB; and therefore the triangle aiic equiangular to the triangle ABC. Wherefore the angle anc or FnD is equal to the angle ABC, and conse- < uently to the angle F/>D ; and there fore the point n falls on the point b, Moreover the angle GPQ, which is half the angle GPD at the centre, is equal to the angle GaD at the circumference \ and the angle GQP, which is half the angle GQD at the centre, is equal to the complement to two right angles of the angle GbD at the circum ference, and therefore equal to the angle Gba. Upon which account the triangles GPQ, Gab, are similar, and Ga is to ab as GP to PQ. ; that is (by construction), as Ga to AB. Wherefore ab and AB are equal; and consequently the triangles abc, ABC, which we have now proved to be similar, are also equal. And therefore since the angles I), E, F, of the triangle DEF do respectively touch the sides ab, ar, be of the triangle afjc/ the figure AECdef may be completed similar and equal to the figure afrcDEFj and by completing it the Problem will be solved. Q.E.F. COR. Hence a right line may be drawn whose parts given in length may be intercepted between three right lines given by position. Suppose the triangle DEF, by the access of its point D to the side EF, arid by having the sides DE, DF placed i>t directum to be changed into a right line whose given part DE is to be interposed between the right lines AB; AC given by position; and its given part DF is to be interposed between the right lines AB; BC, given by position; then, by applying the preceding construction to this case, the Problem will be solved. THE MATHEMATICAL PRINCIPLES [BOOK 1. PROPOSITION XXVIII. PROBLEM XX. To describe a trajectory giren both in kind and magnitude, given parts of which shall be interposed between three right lines given by position. Suppose a trajectory is to be described that may be similar and equal to the curve line DEF, -and may be cut by three right lines AB, AC, BC, given by position, into parts DE and EF, similar and equal to the given parts of this curve line. Draw the right lines DE, EF, DF: and place the angles D, E, F, of this triangle DEF, so as to touch those right lines given by position (by Lem. XXVI). Then about the triangle describe the trajectory, similar and equal to the curve DEF. Q.E.F. LEMMA XXVII. To describe a trapezium given in kind, the angles whereof may be », placed, in respect of four right lines given by position, that are neither all paralhl among themselves, nor converge to one common point, ////// the several angles may touch the several lines. Let the four right lines ABC, AD, BD, CE, be given by position ; the first cutting the second in A, the third in B, and the fourth in C • and suppose a trapezium fghi is to be described that may be similar to the trapezium FCHI, and whose angle /, equal to the given angle F, may touch the right line ABC ; and (lie other angles g, h, i, equal to the other given angles, G, H, I, may touch the other lines AD, BD, CE, re spectively. Join FH, and upon FG. FH, FI describe J% as many segments of circles FSG, FTH, FVI, the first of which FSG may be capable of an angle equal to the angle BAD ; the second FTH capable of an angle equal to the angle CBD ; and the third FVI of an angle equal to the angle ACE. Bnrf>, the segments are to be described towards those sides of the lines FG, FH, FI, that the circular order of the letters FSGF may be the same as of the letters BADB, and that the letters FTHF may turn .ibout in the same order as the letters CBDC and the letters FVIF in the game order as the letters ACE A. Complete the segments into entire cir cles, and let P be the centre of the first circle FSG, Q, the centre of the second FTH. Join and produce both ways the line PQ,, and in it take QR in the same ratio to PQ as BC has to AB. But QR is to be taken towards that side of the point Q that the order of the letters P, Q,, R SEC. V.J OF NATURAL PHILOSOPHY. 15] may be the same as of the letters A, B, C ; and about the centre R with the interval RF describe a fourth circle FNc cutting (lie third circle FVI in c. Join Fc1 cut ting the first circle in a, and the second in / . Draw aG, &H, cl, and let the figure ABC/'4f/ii be made similar to the figure w^cFGHI; and the trapezium fghi will be that which was required to be de scribed. For let the two first circles FSG, FTH cut one the other in K ; join PK, Q,K, RK, "K, 6K, cK, and produce QP to L. The angles FaK, F6K, FcK at the circumferences are the halves of the angles FPK, FQK, FRK, at the centres, and therefore equal to LPK, LQ.K, LRK, the halves of those angles. Wherefore the figure PQRK is iquiangular and similar to the figure «6cK, and consequently ab is to be res PQ, to Q,R, that is, as AB to BC. But by construction, the angles ''Air, /B//,/C?', are equal to the angles F«G, F&H, Fcl. And therefore the figure ABCfghi may be completed similar to the figure abcFGHl. vVliich done a trapezium fghi will be constructed similar to the trapezium FGHI, and which by its angles/, g, h, i will touch the right lines ABC, AD, BD, CE. Q.E.F. COR. Hence a right line may be drawn whose parts intercepted in a given order, between four right lines given by position, shall have a given proportion among themselves. Let the angles FGH, GHI, be so far in creased that the right lines FG, GH, HI, may lie in directum ; and by constructing the Problem in this case, a right line fghi will be drawn, whose parts fg, gh, hi, intercepted between the four right lines given by position, AB and AD, AD and BD, BD and CE, will be one to another as the lines FG, GH, HI, and will observe the same order among them selves. But the same thing may be more readily done in this manner. Produce AB to K and BD to L, so as BK may be to AB as HI to GH ; and DL to BD as GI to FG; and join KL meeting the right line CE in i. Produce iL to M, so as LM may be to iL as GH to HI ; then draw MQ, parallel to LB, and meeting the right line AD in g, and join gi cutting AB, BD in f, h ; I M* say, the thing is done. For let MO- cut the right line AB in Q, and AD the right line KL iu II ^52 THE MATHEMATICAL PRINCIPLES [BOOK I. S, arid draw AP parallel to BD, and meeting iL in P, and §-M to Lh (g\ to hi, Mi to Li, GI to HI, AK to BK) and AP to BL, will be in the same ratio. Cut DL in 11, so as DL to RL may be in that same ratio; and be cause ffS to g~M, AS to AP. and DS to DL are proportional; therefore (ex ceqit.o) as gS to LA, so will AS be to BL, and DS to RL ; and mixtly. BL — RL to Lh — BL, as AS — DS to gS — AS. That is, BR is to Eh as AD is to Ag, and therefore as BD to gQ. And alternately BR is to BD as 13/i to g-Q,, or as fh to fg. But by construction the line BL was cut in D and R in the same ratio as the line FI in G and H ; and therefore BR is to BD as FH to FG. Wherefore fh is to fg as FH to FG. Since, therefore, gi to hi likewise is as Mi to Li, that is, as GI to HI, it is manifest that the lines FI, fi, are similarly cut in G and H, g and //.. Q.E.F. In the construction of this Corollary, after the line LK is drawn cutting CE in i, we may produce iE to V, so as EV may be to Ei as FH to HI, arid then draw V/~ parallel to BD. It will come to the same, if about the centre i with an interval IH, we describe a circle cutting BD in X, and produce iX to Y so as iY may be equal to IF, and then draw Yf parallel to BO. Sir Christopher Wren and Dr. Wallis have long ago given other solu tions of this Problem. PROPOSITION XXIX. PROBLEM XXI. To describe a trajectory given in kind, that may be cut by four right lines given by position, into parts given in order, kind, and proportion. Suppose a trajectory is to be described that may be similar to the curve line FGHI, and whose parts, similar and proportional to the parts FG, GH, HI of the other, may be intercepted between the right lines AB and AD, AD, and BD, BD and CE given by po sition, viz., the first between the first pair of those lines, the second between the second, and the third between the third. Draw the right lines FG, GH, HI, FI; and (by Lem. XXVII) describe a trapezium fghi that may be similar to the trapezium FGHI, and whose an gles/, g, h, i, may touch the right lines given by posi tion AB, AD, BD, CE, severally according to their order. And then about bins trapezium describe a trajectory, that trajectory will be similar to the curve line FGHI. SCHOLIUM. This problem may be likewise constructed in the following manner. Joining FG, GH, HI, FI, produce GF to Y, and join FH, IG, and make SEC. VI OF NATURAL PHILOSOPHY. 153 El the angles CAK. DAL equal to the angles PGH, VFH. Let AK, AL meet the right line BD in K and L, and thence draw KM, LN, of which let KM make the angle A KM equal to the angle CHI, and be itself to AK as HI is to GH ; and let LN make the angle ALN equal to the angle FHI, and be itself to AL as HI to FH. But AK, KM. AL, LN are to be drawn towards those sides of the lines AD, AK, AL, that the letters OA.KMC, ALKA, DALND may be carried round in the same order as the letters FGHIF ; and draw MN meeting the right v line CE in L Make the angle iEP equal to the angle IGF, and let PE be to Ei as FG to GI ; and through P draw PQ/' that may with the right line ADE contain an angle PQE equal to the angle FIG, and may meet the right line AB in /, and join fi. But PE and PQ arc- to be drawn towards those sides of the lines CE, PE, that the circular order of the letters PEtP and PEQP may be the same as of the letters FGHIF ; and if upon the line/i, in the same order of letters, and similar to the trapezium FGHI, a trapezium /^//.i is constructed, and a trajectory given in kind is circumscribed about it, the Problem will be solved. So far concerning the finding of the orbits. It remains that we deter mine the motions of bodies in the orbits so found. SECTION VI. How the motions are to be found in given, orbits. PROPOSITION XXX. PROBLEM XXII. To find at any assigned time the place of a body moving in, a given parabolic trajectory. Let S be the focus, and A the principal vertex of the parabola; and suppose 4AS X M equal to the parabolic area to be cut off APS, which either was described by the radius SP, since the body's departure from the vertex, or is to be described thereby before its arrival there. Now the quantity of that area to be cut off is known from the time which is propor tional to it. Bisect AS in G, and erect the perpendicular GH equal to 3M, and a circle described about th j centre H, with the interval HS, will cut the parabola in the place P required. For letting fall PO perpendic ular on the axis, and drawing PH, there will be AG2 -f- GH2 (=.= HP2 -_ AO^TAGJ* + PO — GH|2) = AO2 + PO2 — 2CA > — ?G!I f PO ' A G S 154 THE MATHEMATICAL PRINCIPLES [BOOK I AG* + GH2. Whence 2GH X PO (« AO2 + PO2 — 2GAO) = AOJ PO2 -f | PO2. For AO2 write AO X'; then dividing all the terms by 2PO; and multiplying them by 2AS, we shall have ^GH X AS (= IAO the area APO — SPO)| = to the area APS. But GH was 3M, and therefore ^GH X AS is 4AS X M. Wherefore the area cut off APS is equal to the area that was to be cut off 4AS X M. Q.E.D. Con. 1. Hence GH is to AS as the time in which the body described the arc AP to the time in which the body described the arc between the vertex A and the perpendicular erected from the focus S upon the axis. COR. 2. And supposing a circle ASP perpetually to pass through the moving body P, the xelocity of the point H is to the velocity which the body had in the vertex A as 3 to 8; and therefore in the same ratio is the line GH to the right line which the body, in the time of its moving from A to P, would describe with that velocity which it had in the ver tex A. COR. 3. Hence, also, on the other hand, the time may be found in which the body has described any assigned arc AP. Join AP, and on its middle point erect a perpendicular meeting the right line GH in H, LEMMA XXVIII. There is no oval figure whose area, cut off by right lines at pleasure, can, be universally found by means of equations of any number of finite terms and dimensions. Suppose that within the oval any point is given, about which as a pole a right line is perpetually revolving with an uniform motion, while in that right line a mov cable point going out from the pole moves always forward with a velocity proportional to the square of that right line with in the oval. By this motion that point will describe a spiral with infinite circumgyrations. Now if a portion of the area of the oval cut off by that right line could be found by a finite equation, the distance of the point from the pole, which is proportional to this area, might be found by the same equation, and therefore all the points of the spiral might be found by a finite equation also ; and therefore the intersection of a right line given in position with the spiral might also be found by a finite equation. But every right line infinitely produced cuts a spiral in an infinite num ber of points ; and the equation by which any one intersection of two lines is found at the same time exhibits all their intersections by as many roots, and therefore rises to as many dimensions as there are intersections. Be cause two circles mutually cut one another in two points, one of those in- 8FC. Vl.J OF NATURAL PHILOSOPHY. 155 terscctions is not to be found but by an equation of two dimensions, fo which the other intersection may be also found. Because there may b(- four intersections of two conic sections, any one of them is not to be found universally, but by an equation of four dimensions, by which they may bi> all found together. For if those intersections are severally sought, be cause the law and condition of all is the same, the calculus will be the same in every case, and therefore the conclusion always the same, which must therefore comprehend all those intersections at once within itself, and exhibit them all indifferently. Hence it is that the intersections of the conic se"f ions with the curves of the third order, because they may amount to six, (\,me out together by equations of six dimensions ; and the inter sections of two curves of the third order, because they may amount to nine, come out together by equations of nine dimensions. If this did not ne cessarily happen, we might reduce all solid to plane Problems, and those higher than solid to solid Problems. But here i speak of curves irreduci ble in power. For if the equation by which the curve is defined may bo reduced to a lower power, the curve will not be one single curve, but com posed of two, or more, whose intersections may be severally found by different calculusses. After the same manner the two intersections of right lines with the conic sections come out always by equations of two dimensions ; the three intersections of right lines with the irreducible curves of the third urder by equations of three dimensions ; the four intersections of right lines with the irreducible curves of the fourth order, by equations of four dimensions ; and so on in iitfinitum. Wherefore the innumerable inter sections of a right line with a spiral, since this is but one simple curve and not reducible to more curves, require equations infinite in r- .imber of dimensions and roots, by which they may be all exhibited together. For the law and calculus of all is the same. For if a perpendicular is let fall from the pole upon that intersecting right line, and that perpendicular together with the intersecting line revolves about the pole, the intersec tions of the spiral will mutually pass the one into the other ; and that which was first or nearest, after one revolution, will be the second ; after two, the third ; and so on : nor will the equation in the mean time be changed but as the magnitudes of those quantities are changed, by which the position of the intersecting line is determined. Wherefore since those quantities after every revolution return to their first magnitudes, the equa tion will return to its first form ; and consequently one and the same equation will exhibit all the intersections, and will therefore have an infi nite number of roots, by which they may be all exhibited. And therefore the intersection of a right line with a spiral cannot be universally found by any finite equation ; and of consequence there is no oval figure whose area, cut off by right lines at pleasure, can be universally exhibited by an^ such equation. 1 56 THE MATHEMATICAL PRINCIPLES [BOOK 1 By the same argument, if the interval of the pole and point by which the spiral is described is taken proportional to that part of the perimeter of the oval which is cut off, it may be proved that the length of the peri meter cannot be universally exhibited by any finite equation. But here I speak of ovals that are not touched by conjugate figures running out in infinitvm. COR. Hence the area of an ellipsis, described by a radius drawn from the focus to the moving body, is not to be found from the time given by a finite equation ; and therefore cannot be determined by the description ol curves geometrically rational. Those curves I call geometrically rational, all the points whereof may be determined by lengths that are definable by equations ; that is, by the complicated ratios of lengths. Other curves (such as spirals, quadratrixes, and cycloids) I call geometrically irrational. For the lengths which are or are not as number to number (according to the tenth Book of Elements) are arithmetically rational or irrational. And therefore I cut off an area of an ellipsis proportional to the time in which it is described by a curve geometrically irrational, in the following manner. PROPOSITION XXXI. PROBLEM XXIII. To find the place of a body moving in a given elliptic trajectory at any assigned time. Suppose A to be the principal vertex, S the focus, and O the centre of the ellipsis A PB ; and let P be the place of the body to be found. Produce OA to G so as OG may be to OA as OA to OS. Erect the perpendicular GH; and about the centre O, with the interval OG, de scribe the circle* GEF ; and on the ruler GH, as a base, suppose the wheel GEF to move forwards, revolving about its axis, and in the mean time by its point A describing the cycloid ALL Which done, take GK to the perimeter GEFG of the wheel, in the ratio of the time in which the body proceeding from A described the arc AP, to the time of a whole revolution in the ellipsis. Erect the perpendicular KL meeting the cycloid in L ; then LP drawn parallel to KG will meet the ellipsis in P, the required place of the body. For about the centre O with the interval OA describe the semi-circle AQB, and let LP, produced, if need be, meet the arc AQ, in Q, and join SEC. VI. OF NATURAL PHILOSOPHY. 157 SQ, OQ. Let OQ meet the arc EFG in F, and upon OQ let fall the perpendicular Sll. The area APS is as the area AQS, that is, as tlie difference between the sector OQA and the triangle OQS, or as the difLi- ence of the rectangles *OQ, X AQ, and -J.OQ X SR, that is, because ' . >,_ is given, as the difference between the arc AQ and the right line Sll : ai.;l therefore (because of the equality of the given ratios SR to the sine of the arc AQ,, OS to OA, OA to OG, AQ to GF; and by division, AQ— Sii to GF — sine of the arc AQ) as GK, the difference between the arc C 1 and tlie sine of the arc AQ. Q.E.D. SCHOLIUM. But since the description of this curve is difficult, a solution by approximation will be preferable. First, then, let there be found a certain angle B which may be to an angle of 57,29578 degrees, which an arc equal to the radius subtends, as SH, the distance of the foci, to AB, the diameter of the ellipsis. Secondly, a certain length L, which may be to the radius in the same ratio inversely. And these being found, the Problem may be solved by the following analysis. By any construction (or even by conjecture), suppose we know P the place of the body near its true place jo. Then letting fall on the axis of the ellipsis the ordinate PR from the proportion of the diameters of the ellipsis, the ordinate RQ of the circumscribed circle AQB will be given ; which ordinate is the sine of the angle AOQ, supposing AO to be the radius, and also cuts the ellipsis in P. It will .be sufficient if that angle is found by a rude calculus in numbers near the truth. Suppose we also know the angle proportional to the time, that is, which is to four right a iules as the time in which tlie body described the arc A/?, to the time of one revolution in the ellipsis. Let this angle be N. Then take an angle D, which may be to the angle B as the sine of the angle AOQ to the radius ; and an angle E which may be to the angle N — AOQ -fD as the length L to the same length L diminished by the cosine of the angle AOQ, when that angle is less than a right angle, or increased thereby when greater. In the next place, take an angle F that may be to the angle B as the sine of the angle 1OQ H- E to the radius, and an angle G, that may be to the angle N- AOQ — E -f F as the length L to the same length L diminished by the cosine of the angle AOQ + E, when that angle is less than a right angle, or increased thereby when greater. For the third time take an angle H, that may be to the angle B as the sine of the angle AOQ f- E 4- G to the radius; and an angle I to the angle N — AOQ — E — G -f- H, as the 58 THE MATHEMATICAL PRINCIPLES jB(OK 1. length L is to the same length L diminished by the cosine of the angle AOQ -f- E + G, when that angle is less than a right angle, or increased thereby when greater. And so we may proceed in infinitum. Lastly, take the angle AOy equal to the angle AOQ -f- E 4- G + I -\-} &c. and from its cosine Or and the ordinatejor, which is to its sine qr as the lesser axis of the ellipsis to the greater, \\ e shall have p the correct place of the body. When the angle N — AOQ, -f D happens to be negative, the sign -|- of the angle E must be every where changed into — , and the sign — into +. And the same thing is to be understood of the signs of the angles G and I, when the angles N — AOQ — E -f F, and N — AOQ — E — G + H come out negative. But the infinite series AOQ -f- E -f- G -|- I +, &c. converges so very fast, that it will be scarcely ever needful to pro ceed beyond the second term E. And the calculus is founded upon this Theorem, that the area APS is as the difference between the arc AQ and the right line let fall from the focus S perpendicularly upon the radius OQ. And by a calculus not unlike, the Problem is solved in the hyperbola. Let its centre be O, its vertex A, its focus S, and asymptote OK ; and suppose the quantity of the area to be cut off is known, as being proportional to the time. Let that be A, and by conjecture suppose we know the position of a rij;ht i ne SP, that cuts off an area APS near the truth. Join OP, and from A and P to the asymptote ° T A S draw AI, PK parallel to the other asymptote ; and by the table of loga rithms the area AIKP will be given, and equal thereto the area OPA, which subducted from the triangle OPS, will leave the area cut off APS. And by applying 2 APS — 2 A, or 2 A — 2 A PS, the double difference of the area A that was to be cut off, and the area APS that is cut off, to the line SN that is let fall from the focus S, perpendicular upon the tangent TP, we shall have the length of the chord PQ. Which chord PQ is to be inscribed between A and P, if the area APS that is cut off be greater than the area A that was to be cut off, but towards the contrary side of the point P, if otherwise : and the point Q will be the place of the body more accurately. And by repeating the computation the place may be found perpetually to greater and greater accuracy. And by such computations we have a general analytical resolution of the Problem. But the par ticular calculus that follows is better fitted for as tronomical purposes. Supposing AO, OB, OD, to be the semi-axis of the ellipsis, and L its latus rec tum, and D the difference betwixt the lesser semi- SEC. VII.] OF NATURAL PHILOSOPHY. J 59 axis OD, and -,L the half of the latus rectum : let an angle Y be found, whose sine may be to the radius as the rectangle under that difference J), and AO 4- OD the half sum of the axes to the square of the greater axis AB. Find also an angle Z, whose sine may be to the radius as the double rec tangle under the distance of the foci SH and that difference D to triple the square of half the greater semi-axis AO. Those angles being once found, the place of the body may be thus determined. Take the angle T proportional to the time in which the arc BP was described, or equal to what is called the mean motion ; and an angle V the first equation of thr mean motion to the angle Y, the greatest first equation, as the sine of double the angle T is to the radius ; and an angle X, the second equation, to the angle Z, the second greatest equation, as the cube of the sine of the angle T is to the cube of the radius. Then take the angle BHP the mean motion equated equal to T + X + V, the sum of the angles T, V. X, if the angle T is less than a right angle; or equal to T + X — V, the difference of the same, if that angle T is greater than one and less than two right angles ; and if HP meets the ellipsis in P, draw SP, and it will cut off the area BSP nearly proportional to the time. This practice seems to be expeditious enough, because the angles V and X, taken in second minutes, if you please, being very small, it will be suf ficient to find two or three of their first figures. But it is likewise sufficiently accurate to answer to the theory of the planet's motions. For even in the orbit of Mars, where the greatest equation of the centre amounts to ten degrees, the error will scarcely exceed one second. But when the angle of the mean motion equated BHP is found, the angle oi the true motion BSP, and the distance SP, are readily had by the known methods. And so far concerning the motion of bodies in curve lines. But it mav also come to pass that a moving body shall ascend or descend in a right line : and I shall now go on to explain what belongs to such kind of motions. SECTION VII. Concerning the rectilinear ascent and descent of bodies, PROPOSITION XXXII. PROBLEM XXIV. Supposing that the centripetal force is reciprocally proportional to tht square of tlie distance of the places from the centre ; it is required to define the spaces which a body, falling directly, describes in given times. CASE 1. If the body does not fall perpendicularly, it will (by Cor. I 160 THE MATHEMATICAL PRINCIPLES [BOOK I Prop. XIII) describe some conic section whose focus is A placed in the centre of force. Suppose that conic sec tion to be A RPB and its focus S. And, first, if the figure be an ellipsis, upon the greater axis thereof AB describe the semi-circle ADB, and let the right line I) PC pass through the falling body, making right angles with the axis; and drawing DS, PS, the area ASD will c be proportional to the area ASP, and therefore also to the time. The axis AB still reaiaining the same, let the breadth of the ellipsis be perpetually diminished, and s the area ASD will always remain proportional to the time. Suppose that breadth to be diminished in, in fruit um ; and the orbit APB in that case coinciding with the axis AB, and the focus S with the extreme point of the axis B, the body will descend in the right line AC1. and the area ABD will become proportional to the time. Wherefore the space AC will be given which the body describes in a given time by its- perpendicular fall from the place A, if the area ABD is taken proportional to the time, and from the point D the right line DC is let fall perpendic ularly on the right line AB. Q,.E.I. CASE 2. If the figure RPB is an hyperbola, on the same principal diameter AB describe the rectangular hyperbola BED ; and because the areas CSP, CB/P, SPy 13, are severally to the sev eral areas CSD, CBED, SDEB, in the given ratio of the heights CP, CD, and the area SP/B is proportional to the time in which the body P will move through the arc P/B. the area SDEB will be also proportional to that time. Let the latus rectum of the hyperbola RPB be diminished in infitiitum, the latus transversum remaining the same; and the arc PB will come to coincide with the right line CB, and the focus S, with the vertex B, A- and the right line SD with the right line BD. And therefore the area BDEB will be proportional to the time in which the body C, by its per pendicular descent, describes the line CB. Q.E.I. CASE 3. And by the like argument, if the figure RPB is a parabola, and to the same principal ver tex B another parabola BED is described, that may always remain given while the former para bola in whose perimeter the body P moves, by having its latus rectum diminished and reduced to nothing, comes to coincide with the line CB, the parabolic segment BDEB will be proportional if to the time in which that body P or C will descend to the centre S or B Q.K.T fl.l OF NATURAL PHILOSOPHY. PROPOSITION XXXIII. THEOREM IX. The tilings above found being supposed. I say, thai the, velocity of a Jai ling body in any place C is to the velocity of a body, describing a circle about the centre B at the distance BC, in, the subduplicate ratio of AC, the distance of the body from the remoter vertex A of the circle or rectangular hyperbola, to iAB, the principal semi-diameter of the Let AB, the common dia meter of both figures RPB, DEB, be bisected in O; and draw the right line PT that may touch the figure RPB in P, and likewise cut that common diameter AB (pro duced, if need be) in T; and let SY be perpendicular to this line, and BQ to this di ameter, and suppose the latus rectum of the figure RPB to be L. Prom Cor. 9, Prop. XVI, it is manifest that the velocity of a body, moving in the line RPB about the centre S, in any place P, is to the velocity of a body describing a circle about the same centre, at the distance SP, in the subduplicate ratio of the rectangle £L X SP to SY2 Por by the properties of the conic sections ACB is to CP2 as 2AO to L. 2CP5 X AO and therefore — — rrrr; is equal to L. Therefore those, velocities an o-- ACB to each other in the subduplicate ratio of CP3 X AO X SP ACB toSY~. More over, by the properties of the conic sections, CO is to BO as BO to TV.? and (by composition or division) as CB to BT. Whence (by division cs composition) BO — or + CO will be to BO as CT to BT, that is, AC CP2 X AO X SP ACB" is equal to will be to AO as CP to BQ; and therefore ~AO X BC * ^°W suPPose GV, tne breadth of the figure RPB, to be diminished in infinitum, so as the point P may come to coincide with the point C, and the point S with the point B. and the line SP with the line BC, and the line SY with the line BQ; and the velocity of the body now descending perpendicularly in the line CB will be to the velocity of 11 162 THE MATHEMATICAL PRINCIPLES [BOOK 1 a body describing a circle about the centre B, at the distance BC, in thr BQ2 X AC X SP subduplicate ratio of - -r-^ - ^ - to SY2, that is (neglecting the ra- X Jo tios of equality of SP to BC, and BQ,2 to SY2), in the subduplicate ratio of AC to AO, or iAB. Q.E.D. COR. 1 . When the points B and S come to coincide, TC will become to TS as AC to AO. COR. 2. A body revolving in any circle at a given distance from the centre, by its motion converted upwards, will ascend to double its distance from the centre. • PROPOSITION XXXIV. THEOREM X. If the. figure BED is a parabola, I say, that the velocity of a falling body in any place C is equal to the velocity by which a body may uniformly describe a circle about the centre B at half the interval BC For (by Cor. 7, Prop. XVI) the velocity of a body describing a parabola RPB about the cen tre S, in any place P, is equal to the velocity of a body uniformly describing a circle about the c same centre S at half the interval SP. Let the breadth CP of the parabola be diminished in itifiiiitirni, so as the parabolic arc P/B may come to coincide with the right line CB, the centre S s with the vertex B, and the interval SP with the interval BC, and the proposition will be manifest. Q.E.D. PROPOSITION XXXV. THEOREM XL The same things supposed, I say, that the area of the figure DES, de scribed by the indefinite radius SD, is equal to the area which a body with a radius equal to h'df the latus rectum of the figure DES, by uniformly revolving about the centre S, may describe in the same tijiw. 1 JD/ AJ SEC. ni: OF NATURAL PHILOSOPHY. For suppose a body C in the smallest moment of time describes in fal ling the infinitely little line Cc. while another body K, uniformly revolv ing about the centre S in the circle OK/r, describes the arc KA:. Erect the perpendiculars CD, cd, meeting the figure DES in D, d. Join SD, Sf/. SK. SA*; and draw Del meeting the axis AS in T, and thereon let fall the perpendicular SY. CASE 1. If the figure DES is a circle, or a rectangular hyperbola, bisect its transverse diameter AS in O, and SO will be half the latus rectum. And because TC is to TD as Cc to Dd, and TD to TS as CD to S Y ; ex aquo TC will be to TS as CD X Cc to SY X Dd. But (by Cor. 1, Prop. XXXIII) TC is to TS as AC to AO; to wit, if in the coalescence of the points D, d, the ultimate ratios of the lines are taken. Wherefore AC is to AO or SK as CD X Cc to S Y X Vd, Farther, the velocity of the descending body in C IF, to the velocity of a body describing a circle about the centre S, at the interval SC, in the subduplicate ratio of AC to AO or SK (by Pi-op. XXXIII) ; and this velocity is to the velocity of a body describing the circle OKA: in the subduplicate ratio of SK to SC (by Cor. 6, Prop IV) ; and, ex aqnnj the first velocity to the last, that is, the little line Cc to the arc K/r, in the subduplicate ratio of AC to SC, that is, in the ratio of AC to CD. Wherefore CD X Cc is equal to AC X KA*, and consequently AC to SK as AC X KA: to SY X IW. and thence SK X KA: equal to SY X Drf, and iSK X KA: equal to £SY X DC/, that is, the area KSA* equal to the area SDrf. Therefore in every moment of time two equal particles, KSA" and SDrf, of areas are generated, which, if their magnitude is diminished, and their number increased in iiifinif't-w, obtain the ratio cf equality, and consequently (by Cor. Lem. IV), the whole areas together generated are always equal. Q..E.D. CASE 2. But if the figure DES is a parabola, we shall find, as above. CD X Cc to SY X Df/ as TC to TS, that is, as 2 to 1 ; and that therefore |CD X Cc is equal to i SY X Vd. But the veloc ity of the falling body in C is equal to the velocity writh which a circle may be uniformly described at the interval 4SC (by Prop" XXXIV). And this velocity to the velocity with which a circle may be described with the radius SK, that is, the little line Cc to the arc KA', is (by Cor. 6, Prop. IV) in the subduplicate ratio of SK to iSC ; that is, in the ratio of SK to *CD. Wherefore iSK X KA: is equal to 4CD X Cc, and therefore equal to £SY X T)d ; that is, the area KSA* is equal to the area SIW, as above. Q.E.D. 164 THE MATHEMATICAL PRINCIPLES [BOOK 1. PROPOSITION XXXVI. PROBLEM XXV. To determine the times of the descent of a body falling from place A. Upon the diameter AS, the distance of the body from the centre at the beginning, describe the semi-circle ADS, as likewise the semi-circle OKH equal thereto, about the centre S. From any place C of the body erect the ordinate CD. ° Join SD, and make the sector OSK equal to the area ASD. It is evident (by Prop. XXXV) that the body in falling will describe the space AC in the same time in which another body, uniformly revolving about the centre S, may describe the arc OK. Q.E.F. M a given PROPOSITION XXXVII. PROBLEM XXVI. To define the times of the ascent or descent of a body projected upwards or downwards from a given place. Suppose the body to go oif from the given place G, in the direction of the line GS, with any velocity. In the duplicate ratio of this velocity to the uniform velocity in a circle, with which the body may revolve about \ H D the centre S at the given interval SG, take GA to £AS. If that ratio is the same as of the number 2 to 1, the point A is infinitely remote ; in which case a parabola is to be described with any latus rectum to the ver tex S, and axis SG ; as appears by Prop. XXXIV. But if that ratio is less or greater than the ratio of 2 to 1, in the former case a circle, in the latter a rectangular hyperbola, is to be described on the diameter SA; as appears by Prop. XXXIII. Then about the centre S, with an interval equal to half the latus rectum, describe the circle H/vK ; and at the place G of the ascending or descending body, and at any other place C, erect the perpendiculars GI, CD, meeting the conic section or circle in I and D. Then joining SI, SD, let the sectors HSK, HS& be made equal to the segments SEIS, SEDS. and (by Prop. XXXV) the body G will describe SEC. VII.] OF NATURAL PHILOSOPHY. 165 the space GO in the same time in which the body K may describe t*he arc Kk. Q.E.F. PROPOSITION XXXVIII. THEOREM XII. Supposing that the centripetal force is proportional to the altitude or distance of places from the centre, I say, that the times and velocities of falling bodies, and the spaces which they describe, are respectively proportional to the arcs, and the right and versed sines of the arcs. Suppose the body to fall from any place A in the A. right line AS ; and about the centre of force S, with the interval AS, describe the quadrant of a circle AE ; and let CD be the right sine of any arc AD ; and the body A will in the time AD in falling describe the space AC, and in the place C will acquire the ve locity CD. This is demonstrated the same way from Prop. X, as Prop. XXX11 was demonstrated from Prop. XI. COR. 1. Hence the times are equal in which one body falling from the place A arrives at the centre S, and another body revolving describes the quadrantal arc ADE. COR. 2. Wherefore all the times are equal in which bodies falling from whatsoever places arrive at the centre. For all the periodic times of re volving bodies are equal (by Cor. 3; Prop. IV). PROPOSITION XXXIX. PROBLEM XXVIT. Supposing a centripetal force of any kind, and granting the quadra- tnres of curvilinear figures ; it is required to find the velocity of a bod)/, ascending or descending in a right line, in the several places through which it passes ; as also the time in which it will arrive at any place : and vice versa. Suppose the body E to fall from any place A in the right line AD EC ; and from its place E imagine a perpendicular EG always erected proportional to the centripetal force in that place tending to the centre C ; and let BFG be a curve line, the locus of the point G. And D in the beginning of the motion suppose EG to coincide with the perpendicular AB ; and the velocity of the body in any place E will be as a right line whose square is equal to the cur vilinear area ABGE. Q.E.I. In EG take EM reciprocally proportional to E 366 THE MATHEMATICAL PRINCIPLES [BOOK 1 a right line whose square is equal to the area ABGE, and let VLM be a curve line wherein the point M is always placed, and to which the right line AB produced is an asymptote; and the time in which the body in falling- describes the line AE, will be as the curvilinear area ABTVME. Q.E.I. For in the right line AE let there be taken the very small line DE of a given length, and let DLF be the place of the line EMG, when the body was in D ; and if the centripetal force be such, that a right line, whose square is equal to the area ABGE; is as the velocity of the descend ing body, the area itself will be as the square of that velocity ; that is, if for the velocities in D and E we write V and V + I, the area ABFD will be as VY, and the area ABGE as YY + 2VI -f II; and by division, the area DFGE as 2 VI -f LI, and therefore ^ will be as -- ^r that is. if we take the first ratios of those quantities when just nascent, the 2YI length DF is as the quantity -|yrr» an(i therefore also as half that quantity 1 X Y • But the time in which the body in falling describes the very line DE, is as that line directly and the velocity Y inversely ; and the force will be as the increment I of the velocity directly and the time inversely ; and therefore if we take the first ratios when those quantities I X V are just nascent, as— -jy==r-. that is, as the length DF. Therefore a force proportional to DF or EG will cause the body to descend with a velocity that is as the right line whose square is equal to the area ABGE. Q.E.D. Moreover, since the time in which a very small line DE of a given length may be described is as the velocity inversely, and therefore also inversely as a right line whose square is equal to the area ABFD ; and since the line DL. and by consequence the nascent area DLME, will be as (he same right line inversely, the time will be as the area DLME, and the sum of all the times will be as the sum of all the areas : that is (by Cor. Lern. IV), the whole time in which the line AE is described will be as the whole area ATYME. Q.E.D. COR. 1. Let P be the place from whence a body ought to fall, so as that, when urged by any known uniform centripetal force (such as gravity is vulgarly supposed to be), it may acquire in the place D a velocity equal to the velocity which another body, falling by any force whatever, hath acquired in that place D. In the perpendicular DF let there be taken DR., which may be o DF as that uniform force to the other force in the place D. Complete the rectangle PDRQ,, and cut iff the area. ABFD equal to that rectangle. Then A will be the place SEC. VII. I OF NATURAL PHILOSOPHY. 10; from whence the other body fell. For com pleting the rectangle DRSE, since the area ABFD is to the area DFGE as VV to 2VI, and therefore as 4V to I, that is, as half the whole velocity to the increment of the velocity of the body falling by the unequable force ; and in like manner the area PQRD to the area DRSE as half the whole velocity to the incre ment of the velocity of the body falling by the uniform force ; and since those increments (by reason of the equality of the nascent times) are as the generating forces, that is, as the or- dinates DF, DR, and consequently as the nascent areas DFGE, DRSE : therefore, ex aq-uo, the whole areas ABFD, PQRD will be to one another as the halves of the whole velocities ; and therefore, because the velocities are equal, they become equal also. COR. 2. Whence if any body be projected either upwards or downwards with a given velocity from any place D, and there be given the law of centripetal force acting on it, its velocity will be found in any other place, as e, by erecting the ordinate eg, and taking that velocity to the velocity in the place D as a right line whose square is equal to the rectangle PQRD, either increased by the curvilinear area DFge, if the place e is below the place D, or diminished by the same area DFg-e, if it be higher, is to the right line whose square is equal to the rectangle PQRD alone. COR. 3. The time is also known by erecting the ordinate em recipro cally proportional to the square root of PQRD -f- or — T)Fge, and taking the time in which the body has described the line De to the time in which another body has fallen with an uniform force from P, and in falling ar rived at D in the proportion of the curvilinear area DLme to the rectan gle 2PD X DL. For the time in which a body falling with an uniform force hath described the line PD, is to the time in which the same body has described the line PE in the subduplicate ratio of PD to PE ; that is (the very small line DE being just nascent), in the ratio of PD to PD -f ^DE; or 2PD to 2PD -f- DE, and, by division, to the time in which the body hath described the small line DE, as 2PD to DE, and therefore as the rectangle 2PD X DL to the area DLME ; and the time in which both the bodies described the very small line DE is to the time in which the body moving unequably hath described the line De as the area DLME to the area DLme ; and, ex aquo, the first mentioned of these times is to the last as the rectangle 2PD X DL to the area DLrae. 163 THE MATHEMATICAL PRINCIPLES [BoOK I SECTION VIII. Of the invention of orbits wherein bodies will revolve, being acted upon by any sort of centripetal force. PROPOSITION XL. THEOREM XIII. //' a body, acted upon by any centripetal force, is any how moved, and another body ascends or descends in a right line, and their velocities be equal in amj one case of equal altitudes, t/ieir velocities will be also equal at all equal altitudes. Let a body descend from A through D and E, to the centre (j : and let another body move from V in the curve line VIK&. From the centre C, with any distances, describe the concentric circles DI, EK, meeting the right line AC in I) and E; and the curve VIK in I and K. Draw 1C meeting KE in N, and on IK let fall the perpendicular NT • and let the interval DE or IN between the circumferences of the circles be very small ; K/ and imagine the bodies in D and I to have equal velocities. Then because the distances CD and CI are equal, the centri petal forces in D and I will be also equal. Let those forces be k) expressed by the equal lineoke DE and IN ; and let the force IN (by Cor. 2 of the Laws of Motion) be resolved into two others, NT and IT. rl hen the force NT acting in the direction line NT perpendicular to the path ITK of the body will not at all affect or change the velocity of the body in that path, but only draw it aside from a rectilinear course, and make it deflect perpetually from the tangent of the orbit, and proceed in the curvilinear path ITK/j. That whole force, therefore, will be spent in producing this effect: but the other force IT, acting in the direction of the course of the body, will be all employed in accelerating it, and in the least given time will produce an acceleration proportional to itself. Therefore the accelerations of the bodies in D and I, produced in equal times, are as the lines DE, IT (if we take the first ratios of the nascent lines DE, IN, IK, IT, NT) ; and in unequal times as those lines and the times conjunctly. But the times in which DE and IK are described, are, by reason of the equal velocities (in D and I) as the spaces described DE and IK, and therefore the accelerations in the course of the bodies through the lines DE and IK are as DE and IT, and DE and IK conjunctly ; that is, as the square of DE to the rectangle IT into IK. But the rectangle IT X IK is equal to the square of IN, that is, equal to the square of DE ; and therefore the accelerations generated in the passage of the bodies from D and I to E and K are equal. Therefore the velocities of the holies in E and K are also equal, and by the same reasoning they will always be found equal in any subsequent equal dis tances. Q..E.D. SEC. VI11.J OF NATURAL PHILOSOPHY. 169 By the same reasoning, bodies of equal velocities and equal distances from the centre will be equally retarded in their ascent to equal distances. Q.E.D. COR. 1. Therefore if a body either oscillates by hanging to a string, or by any polished and perfectly smooth impediment is forced to move in a curve line ; and another body ascends or descends in a right line, and their velocities be equal at any one equal altitude, their velocities will be also equal at all other equal altitudes. For by the string of the pendulous body, or by the impediment of a vessel perfectly smooth, the same thing will be effected as by the transverse force NT. The body is neither accelerated nor retarded by it, but only is obliged to leave its rectilinear course. COR. 2. Suppose the quantity P to be the greatest distance from the centre to which a body can ascend, whether it be oscillating, or revolving in a trajectory, and so the same projected upwards from any point of a trajectory with the velocity it has in that point. Let the quantity A be the distance of the body from the centre in any other point of the orbit ; and let the centripetal force be always as the power An — ', of the quantity A, the index of which power n — 1 is any number n diminished by unity. Then the velocity in every altitude A will be as v/ P11 — A", and therefore will be given. For by Prop. XXXIX, the velocity of a body ascending and descending in a right line is in tha't very ratio. PROPOSITION XLI. PROBLEM XXVTII. Supposing a centripetal force of any kind, and granting the quadra tures of curvilinear figures, it is required to find as well the trajecto ries in which bodies will move, as the times of their motions in the trajectories found. Let any centripetal force tend to the centre C, and let it be required to find the trajectory VIKAr. Let R, there be given the circle VR, described from the centre C with any interval CV; and from the same centre de scribe any other circles ID, KE cut ting the trajectory in I and K, and the right line CV in D and E. Then draw the right line CNIX cutting the c circles KE, VR in N and X, and the right line CKY meeting the circle VJi in Y. Let the points I and K be indefinitely near ; and let the body go on from V through I and K to k ; and let the point A be the place from whence anothe body is to fall, so as in the place D to acquire a ve locity equal to the velocity of the first body in I. And things remaining as in Prop. XXXIX, the lineola IK, described in the least given time THE MATHEMATICAL PRINCIPLES [BOOK 1 trill be as the velocity, and therefore as the right line whose square is equal to the area ABFD, and the triangle ICK proportional to the time will be given, and therefore KN will be reciprocally as the altitude 1C : that is (if there be given any quantity Q, and the altitude 1C be called A), as -T-. This quantity — call Z, and suppose the magnitude of Q, to oe such that in some case v/ABFD may be to Z as IK to KN, and then in all cases V ABFD will be to Z as IK to KN, and ABFD to ZZ as IK2 to KN2, and by division ABFD — ZZ to ZZ as IN2 to KN2, and there- fore V ABFD — ZZ to Z, or — as IN to KN; and therefore A x KN Q. x IN \vill be equal to — — — . Therefore since YX X XC is to A X KN — ZZ Q. X IN x CX2 as CX2, to AA, the rectangle XY X XC will be equal to- AAv/ABFD — ZZ. Therefore in the perpendicular DF let there be taken continually I)//, IV Q ax ex2 equal to — , =. — respectively, and 2 v/ ABFD — ZZ 2AA V ABFD — ZZ let the curve lines ab, ac, the foci of the points b and c, be described : and from the point V let the perpendicular Va be erected to the line AC, cut ting off the curvilinear areas VD&a, VDra, and let the ordi nates Es:? E#, be erected also. Then because the rectangle D& X IN or DbzR is equal to half the rectangle A X KN, or to the triangle ICK ; and the rectangle DC X IN or Dc.rE is equal to half the rectangle YX X XC, or to the triangle XCY; that is, because the nascent particles I)6d3, ICK of the areas VD/>#, VIC are always equal; and the nascent particles Dc^-E, XCY of the areas VDca, VCX are always equal : therefore the generated area VD6a will be equal to the generated area VIC, and there fore proportional to the time; and the generated area VDco- is equal to the generated sector VCX. If, therefore, any time be given during which the body has been moving from V, there will be also given the area pro portional to it VD/>«; and thence will be given the altitude of the body CD or CI ; and the area VDca, and the sector VCX equal there' o, together with its angle VCL But the angb VCI, and the altitude CI being given, there is also given the place I, in which the body will be found at the end of that time. Q.E.I. COR. 1. Hence the greatest and least altitudes of the bodies, that is, the apsides of the trajectories, may be found very readily. For the apsides are those points in which a right line 1C drawn through the centre falls perpendicularly upon the trajectory VTK; which comes to pass when the right lines IK and NK become equal; that is, when the area ABFD ig C'nl to ZZ. OF NATURAL PHILOSOPHY. 171 SEC. VI1LJ COR. 2. So also the angle KIN, in which the trajectory at any place cuts the line 1C. may be readily found by the given altitude 1C of the body : to wit, by making the sine of that angle to radius as KN to IK that is, as Z to the square root of the area ABFD. COR. 3. If to the centre C, and the principal vertex V, there be described a conic section VRS ; and from any point thereof, as R, there be drawn the tangent T RT meeting the axis CV indefinitely pro duced in the point T ; and then joining C CR there be drawn the right line CP, Q- equal to the abscissa CT, making an angle VCP proportional to the sector VCR ; and if a centripetal force, reciprocally proportional to the cubes of the distances of the places from the centre, tends to the centre C ; and from the place V there sets out a body with a just velocity in the direc tion of a line perpendicular to the right, line CV; that body will proceed in a trajectory VPQ,, which the point P will always touch ; and therefore if the conic section VI\ S be an hyberbola, the body will descend to the cen tre ; but if it be an ellipsis, it will ascend perpetually, and go farther and farther off in infinilum. And, on the contrary, if a body endued with any velocity goes off from the place V, and according as it begins either to de scend obliquely to the centre, or ascends obliquely from it, the figure VRS be either an hyperbola or an ellipsis, the trajectory may be found by increas ing or diminishing the angle VCP in a given ratio. And the centripetal force becoming centrifugal, the body will ascend obliquely in the trajectory VPQ, which is found by taking the angle VCP proportional to the elliptic sector VRC, and the length CP equal to the length CT, as before. All these things follow from the foregoing Proposition, by the quadrature of a certain ourve, the invention of which, as being easy enough, for brevity's sake I omit. PROPOSITION XLII. PROBLEM XXIX. The law of centripetal force being given, it is required to find the motion of a body setting out from a given place, with a given velocity, in the direction of a given right line. Suppose the same things as in Ihe three preceding propositions; and let the body go off from the place I in the direction of the little line, IK, with the same ve locity as another body, by falling with an uniform centripetal force from the place P, may acquire in I); and let this uniform force be to the force with which the body 1.72 THE MATHEMATICAL PRINCIPLES [BOOK 1. is at first urged in I, as DR to DF. Let the body go on towards k; and about the centre C, with the interval Ck, describe the circle ke, meeting the right line PD in e, and let there be erected the lines eg, ev, ew, ordi- nately applied to the curves BF§*, abv} acw. From the given rectangle PDRQ, and the given law of centripetal force, by which the first body is acted on, the curve line BF»* is also given, by the construction of Prop. XXVII, and its Cor. 1. Then from the given angle CIK is given the proportion of the nascent lines 1K; KN ; and thence, by the construction of Prob. XXVIII, there is given the quantity Q,, with the curve lines abv, acw ; and therefore, at the end of any time Dbve, there is given both the altitude of the body Ce or Ck, and the area Dcwe, with the sector equal to it XCy, the angle 1C A:, and the place k} in which the body will then be found. Q.E.I. We suppose in these Propositions the centripetal force to vary in its recess from the centre according to some law, which any one may imagine at pleasure; but at equal distances from the centre to be everywhere the Bame. I have hitherto considered the motions of bodies in immovable orbits. It remains now to add something concerning their motions in orbits which revolve round the centres of force. SECTION IX. Of the motion of bodies in moveable orbits ; and of the motion of the apsides. PROPOSITION XLIII. PROBLEM XXX. Ft is required to make a body move in a trajectory that revolves about the centre of force in the same manner as another body in the same trajectory at rest. In. the orbit VPK, given by position, let the body P revolve, proceeding from V towards K. From the centre C let there be continually drawn Cp, equal to CP, making the angle VC/? proportional to the angle VCP ; and the area which the line Cp describes will be to the area VCP, which the line CP describes at the same time, ns the velocity of the describing line Cp to the velocity of the describing line CP ; that is, as the angle VC/? to the angle VCP, therefore in a given ratio, and therefore proportional to the time. Since, then, the area described by the line Cp in an immovable plane is proportional to the time, it is manifest that a body, being acted upon by a just quantity of centripetal force may SEC. L\.] OF NATURAL PHILOSOPHY. 173 revolve with the point p in the curve line which the same point p, by the method just now explained, may be made to describe an immovable plane. Make the angle VC^ equal to the angle PC/?, and the line Cu equal to CV, and the figure uCp equal to the figure VCP; and the body being al ways in the point p} will move in the perimeter of the revolving figure nCp, and will describe its (revolving) arc up in the same time the* the other body P describes the similar and equal arc VP in the quiescov.t fig ure YPK. Find, then, by Cor. 5, Prop. VI., the centripetal force by which the body may be made to revolve in the curve line which the pom* p de scribes in an immovable plane, and the Problem will be solved. O/E.K. PROPOSITION XLIV. THEOREM XIV. The difference of the forces, by which two bodies may be madi, to KMVG equally, one in a quiescent, the other in the same orbit revolving, i 1 in a triplicate ratio of their common altitudes inversely. Let the parts of the quiescent or bit VP, PK be similar and equal to the parts of the revolving orbit up, pk ; and let the distance of the points P and K be supposed of the utmost smallness Let fall a perpendicular kr from the point k to the right line pC, and produce it to m, so that mr may be to kr as the angle VC/? to the /2\- angle VCP. Because the altitudes of the bodies PC and pV, KG and kC} are always equal, it is manifest that the increments or decrements of the lines PC and pC are always equal ; and therefore if each of the several motions of the bodies in the places P and p be resolved into two (by Cor. 2 of the Laws of Motion), one of which is directed towards the centre, or according to the lines PC, pC, and the other, transverse to the former, hath a direction perpendicular to the lines PC and pC ; the mo tions towards the centre will be equal, and the transverse motion of the body p will be to the transverse motion of the body P as the angular mo tion of the line pC to the angular motion of the line PC ; that is, as the angle VC/? to the angle VCP. Therefore, at the same time that the bodv P, by both its motions, comes to the point K, the body p, having an equal motion towards the centre, will be equally moved from p towards C ; arid therefore that time being expired, it will be found somewhere in the line mkr, which, passing through the point k, is perpendicular to the line pC ; and by its transverse motion will acquire a distance from the line 174 THE MATHEMATICAL PRINCIPLES [BOOK J. »C, that will be to the distance which the other body P acquires from the line PC as the transverse motion of the body p to the transverse motion of the other body P. Therefore since kr is equal to the distance which the body P acquires from the line PC, and mr is to kr as the angle VC/? to the angle VCP, that is, as the transverse motion of the body p to the transverse motion of the body P, it is manifest that the body p, at the ex piration of that time, will be found in the place m. These things will be so, if the bodies jo and P are equally moved in the directions of the lines pC and PC, and are therefore urged with equal forces in those directions. I: ut if we take an angle pCn that is to the angle pCk as the angle VGj0 to the angle VCP, and nC be equal to kG, in that case the body p at the expiration of the time will really be in n ; and is therefore urged with a greater force than the body P, if the angle nCp is greater than the angle kCp, that is, if the orbit npk, move either in cmiseqnentia, or in antece- denticij with a celerity greater than the double of that with which the line CP moves in conseqnentia ; and with a less force if the orbit moves slower in antecedent-la. And ihj difference of the forces will be as the interval mn of the places through which the body would be carried by the action of that difference in that given space of time. About the centre C with the interval Cn or Ck suppose a circle described cutting the lines mr, tun pro duced in s and £, and the rectangle mn X nit will be equal to the rectan- *//? n ^* */?? ^ •"•le mk X ins, and therefore mn will be equal to — — . But since mt the triangles pCk, pCn, in a given time, are of a given magnitude, kr and mr. a id their difference mk, and their sum ms, are reciprocally as the al titude pC, and therefore the rectangle mk X ms is reciprocally as the square of the altitude pC. But, moreover, mt is directly as |//z/, that is, as the altitude pC. These are the first ratios of the nascent lines ; and hence — r — - that is, the nascent lineola mn. and the difference of the forces mt proportional thereto, are reciprocally as the cube of the altitude pC. Q.E.D. COR. I. Hence the difference of the forces in the places P and p, or K and /.*, is to the force with which a body may revolve with a circular motion from R to K, in the same time that the body P in an immovable orb de scribes the arc PK, as the nascent line m,n to the versed sine of the nascent mk X ms rk2 arc RK, that is, as — — to ^g, or as mk X ms to the square of rk ; that is. if we take given quantities F and G in the same ratio to one another as the angle VCP bears to the angle VQ?, as GG — FF to FF. And, therefore, if from the centre C, with any distance CP or Cp, there be described a circular sector equal to the whole area VPC, which the body OEC. IX.l OF NATURAL PHILOSOPHY. 175 revolving in an immovable orbit has by a radius drawn to the centre de- bribed in any certain time, the difference of the forces, with which the body P revolves in an immovable orbit, and the body p in a movable or bit, will be to the centripetal force, with which another body by a radius drawn to the centre can uniformly describe that sector in the same time as the area VPC is described, as GG — FF to FF. For that sector and the area pCk are to one another as the times in which they are described. COR. 2. If the orbit YPK be an ellipsis, having its focus C, and its highest apsis Y, and we suppose the the ellipsis upk similar and equal to .. it, so that pC may be always equal / to PC, and the angle YC/? be to the •; angle YCP in the given ratio of G \ to F ; and for the altitude PC or pC \ we put A, and 2R for the latus rec- /t\ turn of the ellipsis, the force with * which a body may be made to re volve in a movable ellipsis will be as FF RGG — RFF — - + - — -rg — — , and vice versa. /Y A. A. Let the force with which a body may revolve in an immovable ellipsis be expressed by the quantity , and the -. 7 force in V will be FF But the force with which a body may revolve in a circle at the distance CY, with the same velocity as a body revolving in an ellipsis has in Y, is to the force with which a body revolving in an ellip sis is acted upon in the apsis Y, as half the latus rectum of the ellipsis to the RFF semi-diameter CY of the circle, and therefore is as , =- : and tlu RFF which is to this, as GG — FF to FF, is as - ~py^~~ ~: and this force (by Cor. 1 cf this Prop.) is the difference of the forces in Y, with which the body P revolves in the immovable ellipsis YPK, and the body p in the movable ellipsis upk. Therefore since by this Prop, that difference at any other altitude A is to itself at the altitude CY as -r-, to ^TF-» the same AJ CYJ R C^ ( "* R P^ T*1 difference in every altitude A will be as - — 3 — : — . Therefore to the FF force -T-: , by which the body may revolve in an immovable ellipsis VPK 176 THE MATHEMATICAL PRINCIPLES [BOOK I. idd the excess — -:-= , and the sum will be the whole force -r-r -\- A AA RGG — RFF, .-5 — by which a body may revolve in the same time in the mot- A. •able ellipsis upk. COR. 3. In the same manner it will be found, that, if the immovable or bit VPK be an ellipsis having its centre in the centre of the forces C} and there be supposed a movable ellipsis -upk, similar, equal, and concentrical to it ; and 2R be the principal latus rectum of that ellipsis, and 2T the latus transversum, or greater axis ; and the angle VCjo be continually to the angle TCP as G to F ; the forces with which bodies may revolve in the im- FFA FFA movable and movable ellipsis, in equal times, will be as ^ and -p™~ RGG — RFF + — .-3 respectively. A COR. 4. And universally, if the greatest altitude CV of the body be called T, and the radius of the curvature which the orbit VPK has in Y, that is, the radius of a circle equally curve, be called R, and the centripetal force with which a body may revolve in any immovable trajectory VPK at the place VFF V be called -f-=Trri , and in other places P be indefinitely styled X ; and the altitude CP be called A, and G be taken to F in the given ratio of the angle VCjD to the angle VCP ; the centripetal force with which the same body will perform the same motions in the same time, in the same trajectory •upk revolving with a circular motion, will be as the sum of the forces X -f- VRGG — VRFF ~ A* COR. 5. Therefore the motion of a body in an immovable orbit being given, its angular motion round the centre of the forces may be increased or diminished in a given ratio; and thence new immovable orbits may be found in which bodies may revolve with new centripetal forces. COR. 6. Therefore if there be erected the line VP of an indeterminate -p length, perpendicular to the line CV given by po sition, and CP be drawn, and Cp equal to it, mak ing the angle VC/? having a given ratio to the an gle VCP, the force with which a body may revolve in the curve line Vjo/r, which the point p is con tinually describing, will be reciprocally as the cube C of the altitude Cp. For the body P, by its vis in ertia alone, no other force impelling it, will proceed uniformly in the right line VP. Add, then, a force tending to the centre C reciprocally as the cube of the altitude CP or Cp, and (by what was just demonstrated) the SEC. IX..J OF NATURAL PHILOSOPHY. 177 body will deflect from the rectilinear motion into the curve line Ypk. But this curve ~Vpk is the same with the curve VPQ found in Cor. 3, Prop XLI, in which, I said, hodies attracted with such forces would ascend obliquely. PROPOSITION XLV. PROBLEM XXXL To find the motion of the apsides in orbits approaching very near to circles. This problem is solved arithmetically by reducing the orbit, which a body revolving in a movable ellipsis (as in Cor. 2 and 3 of the above Prop.) describes in an immovable plane, to the figure of the orbit whose apsides are required ; and then seeking the apsides of the orbit which that body describes in an immovable plane. But orbits acquire the same figure, if the centripetal forces with which they are described, compared between themselves, are made proportional at equal altitudes. Let the point V be the highest apsis, and write T for the greatest altitude CV, A for any other altitude CP or C/?, and X for the difference of the altitudes C V — CP : and the force writh which a body moves in an ellipsis revolving about its •p -p T? C* f ^ T? F* F focus C (as in Cor. 2), and which in Cor. 2 was as -r-r -\ — — -.-3 — — , FFA + RGG — RFF , that is as, -^ — , by substituting T — X for A, will be- A RGG — RFF + TFF — FFX come as -p — . In like manner any other cen tripetal force is to be reduced to a fraction whose denominator is A3, and the numerators are to be made analogous by collating together the homo logous terms. This will be made plainer by Examples. EXAMPLE 1. Let us suppose the centripetal force to be uniform, A3 and therefore as —3 or, writing T — X for A in the numerator, as T3 — 3TTX + 3TXX — X3 — — — =-. Ihen collating together the correspon- A3 dent terms of the numerators, that is, those that consist of given quantities, with those of given quantities, and 'those of quantities not given with those of quantities not given, it will become RGG — RFF -f- TFF to T3 as — FFX to 3TTX -f 3TXX — X3, or as — FF to — 3TT + 3TX — XX. Now since the orbit is supposed extremely near to a circle, let it coincide with a circle ; and because in that case R and T become equal, and X is infinitely diminished, the last ratios will be, as RGG to T2, so — FF to — 3TT, or as GG to TT, so FF to 3TT; and again, as GG to FF, so TT to 3TT, that is, as 1 to 3 ; and therefore G is to F, that is, the angle VC/? to the angle VCP, as 1 to v/3. Therefore since the body, in an immovable 178 THE MATHEMATICAL PRINCIPLES [BOOK I ellipsis, in descending from the upper to the lower apsis, describes an angle, if I may so speak, of ISO deg., the other body in a movable ellipsis, and there fore in the immovable orbit we are treating of, will in its descent from 180 the upper to the lower apsis, describe an angle VCjt? of — ^ deg. And this \/o comes to pass by reason of the likeness of this orbit which a body acted upon by an uniform centripetal force describes, and of that orbit which a body performing its circuits in a revolving ellipsis will describe in a quies cent plane. By this collation of the terms, these orbits are made similar ; not universally, indeed, but then only when they approach very near to a circular figure. A body, therefore revolving with an uniform centripetal 180 force in an orbit nearly circular, will always describe an angle of — » deg/, or v/o 103 deg., 55 m., 23 sec., at the centre; moving from the upper apsis to the lower apsis when it has once described that angle, and thence returning to the upper apsis when it has described that angle again ; and so on in in- finitwn. EXAM. 2. Suppose the centripetal force to be as any power of the alti- An tude A, as, for example, An — 3, or-r^ ; where n — 3 and n signify any in- A. dices of powers whatever, whether integers or fractions, rational or surd, affirmative or negative. That numerator An or T — X|n being reduced to an indeterminate series by my method of converging series, will become Tn — >/XTn— T + • — ^ XXTn— 2, &c. And conferring these terms with the terms of the other numerator RGG — RFF + TFF — FFX, it becomes as RGG — RFF 4- TFF to Tn, so — FF to — ?/.Tn— r + ?~^ XTn — 2, &c. And taking the last ratios where the orbits approach to circles, it becomes as RGG to T'1, so — FF to — nT-1— T, or as GG to T"— ', so FF to ?*Tn— ; and again, GG to FF, so Tn— l to nT"—1, that is, as 1 to n ; and therefore G is to F, that is the angle VCp to the angle VCP, as 1 to ^/n. Therefore since the angle VCP, described in the de scent of the body from the upper apsis to the lower apsis in an ellipsis, is of 180 deg., the angle VC/?, described in the descent of the body from the upper apsis to the lower apsis in an orbit nearly circular which a body de scribes with a centripetal force proportional to the power An — 3, will be equal ISO to an angle of - — deg., and this angle being repeated, the body will re- \/ti turn from the lower to the upper apsis, and so on in infinitum. As if the centripetal force be as the distance of the body from the centre, that is, as A, A4 or -p, n will be equal to 4, and ^/n equal to 2 ; and thereLre the angle IX.] OF NATURAL PHILOSOPHY. IT9 ISO between the upper and the lower apsis will be equal to — deg., or 90 deg. Therefore the body having performed a fourth part of one revolution, will arrive at the lower apsis, and having performed another fourth part, will arrive at the upper apsis, and so on by turns in infiuitum. This appears also from Prop. X. For a body acted on by this centripetal force will re volve in an immovable ellipsis, whose centre is the centre of force. If the 1 A2 centripetal force is reciprocally as the distance, that is, directly as — or — £ ' A A"' ji will be equal to 2 ; and therefore the angle between the upper and lower 180 apsis will be — - deg., or 127 deg., 16 min., 45 sec. ; and therefore a body re v/2 volving with such a force, will by a perpetual repetition of this angle, move alternately from the upper to the lower and from the lower to the upper apsis for ever. So. also, if the centripetal force be reciprocally as the biquadrate root of the eleventh power of the altitude, that is, reciprocally as A —, and, therefore, directly as -r-fp or as Ts> n wil* ^e etlual f° \> an(1 4 A^- A 1 Of) - deg. will be equal to 360 deg. ; and therefore the body parting from v/ n the upper apsis, and from thence perpetually descending, will arrive at the lower apsis when it has completed one entire revolution ; and thence as cending perpetually, when it has completed another entire revolution, it will arrive again at the upper apsis ; and so alternately for ever. EXAM. 3. Taking m and n for any indices of the powers of the alti tude, and b and c for any given numbers, suppose the centripetal force 6Ara + cA" b into T — X> -f- c into T — X to be as r^ that is, as A3 A3 or (by the method of converging series above-mentioned) as bTm + cTn — m6XT" -1 //cXTn— ' mm — m vvrpm un — n ~~2 -- 0A.A1 — ^— — t-XXT" — 2, «fcc. — T$~— ~ and comparing the terms of the numerators, there will arise RGG — II FF -f TFF to ^Tm + cT" as — FF to — mbTm — i — " - « + 2" m bXT" - * + "^p cXTn - » .fee. And tak- ing the last ratios that arise when the orbits come to a circular form, there will come forth GG to 6Tm — l -f cTn — 1 as FF to mbTm — l + ncT" — J ; and again, GG to FF as 6Tm — ' + cTn — ' to mbTn — 1 -f ncTn — \ This proportion, by expressing the greatest altitude CV or T arithmeti cally by unity, becomes, GG to FF as b -{- c to mb -\- ?/c, and therefore as I (80 THE MATHEMATICAL PRINCIPLES [BOOK 1 tub ~h nc to - y—7— — • Whence G becomes to P, that is, the angle VCjo to the an- f) ~T~ C gle VCP. as 1 to >/- . -'-. And therefore since the angle VCP between the upper and the lower apsis, in an immovable ellipsis, is of 180 deg., thr angle VC/? between the same apsides in an orbit which a body describes b A m I c A n with a centripetal force, that is. as - — r§— , will be equal to an angle of A. ISO v/— 1~TT~ ; deg. And °y tne same reasoning, if the centripetal force be as - —73 , the angle between the apsides will be found equal to fi f* 18o V'' — - — - deg. After the same manner the Problem is solved in nib — >ic more difficult cases. The quantity to which the centripetal force is pro portional must always be resolved into a converging series whose denomi nator is A*. Then the given part of the numerator arising from that operation is to be supposed in the same ratio to that part of it which is not given, as the given part of this numerator RGG — RFF -f TFF — FFX. is to that part of the same numerator which is not given. And taking away the superfluous quantities, and writing unity for T, the proportion of G to F is obtained. COR. 1 . Hence if the centripetal force be as any power of the altitude, that power may be found from the motion of the apsides ; and so contra riwise. That is, if the whole angular motion, with which the body returns to the same apsis, be to the angular motion of one revolution, or 360 deg., MS any number as m to another as n, and the altitude called A ; the force nn will be as the power A HSii — 3 of the altitude A; the index of which power is - — 3. This appears by the second example. Hence it is plain that the force in its recess from the centre cannot decrease in a greater than a triplicate ratio of the altitude. A body revolving with such a force, and parting from the apsis, if it once begins to descend, can never arrive at the lower apsis or least altitude, but will descend to the centre, describing the curve line treated of in Cor. 3, Prop. XLL But if it should, at its part ing from the lower apsis, begin to ascend never so little, it will ascend in irtfimtifm, and never come to the upper apsis ; but will describe the curve line spoken of in the same Cor., and Cor. 6, Prop. XLIV. So that where the force in its recess from the centre decreases in a greater than a tripli cate ratio of the altitude, the body at its parting from the apsis, will either descend to the centre, or ascend in iiiftnitum, according as it descends or Ascends at the beginning of its motion. But if the force in its recess from "SEC. IX.J OF NATURAL PHILOSOPHY. ISi the centre either decreases in a less than a triplicate ratio of the altitude, or increases in any ratio of the altitude whatsoever, the body will never descend to the centre, but will at some time arrive at the lower apsis ; and, on the contrary, if the body alternately ascending and descending from one apsis to another never comes to the centre, then either the force increases in the recess from the centre, or it decreases in a less than a triplicate ratio of the altitude; and the sooner the body returns from one apsis to another, the farther is the ratio of the forces from the triplicate ratio. As if the body should return to and from the upper apsis by an alternate descent and ascent in 8 revolutions, or in 4, or 2, or \\ • that is, if m should be to n as 8, or 4, or 2, or H to 1. and therefore --- 3, be g\ — 3,or TV~ 3, or i — 3, or mm 3 I - 3 ; then the force will be as A~ ? or AT° "~ 3j or A*~~ 3j or A"" G ' that is. it will be reciprocally as A3 C4' or A3 T^' or A3 4' or A3 "" If the body after each revolution returns to the same apsis, and the apsis nn _ remains unmoved, then m will be to n as 1 to 1, and therefore A»"« will be equal to A 2, or — - ; and therefore the decrease of the forces will A A be in a duplicate ratio of the altitude ; as was demonstrated above. If the body in three fourth parts, or two thirds, or one third, or one fourth part of an entire revolution, return to the same apsis ; m will be to n as £ or ? n n i_6 _ 3 9 _ 3 o or ^ or i to 1, and therefore Amm 3 is equal to A 9 'or A4 or A _ 3 1 6 _ 3 l_l ' ' or A ; and therefore the force is either reciprocally as A fl or 3 613 A4' or directly as A or A . Lastly if the body in its progress from the upper apsis to the same upper apsis again, goes over one entire revolution and three deg. more, and therefore that apsis in each revolution of the body moves three deg. in consequentia ; then m will be to u as 363 deg. to 360 deg. or as 121 to 120, and therefore Amm will be equal to 2 9_ 5_ 2_ JJ A "' and therefore the centripetal force will be reciprocally as ^T4"6TT> or reciprocally as A2^4^ very nearly. Therefore the centripetal force decreases in a ratio something greater than the duplicate ; but ap proaching 59f times nearer to the duplicate than the triplicate. COR. 2. Hence also if a body, urged by a centripetal force which is re ciprocally as the square of the altitude, revolves in an ellipsis whose focus is in the centre of the forces ; and a new and foreign force should be added to or subducted from this centripetal force, the motion of the apsides arising from that foreign force may (by the third Example) be known ; and so on the contrary. As if the force with which the body revolves in the ellipsis 182 THE MATHEMATICAL PRINCIPLES [BOOK I oe as -r-r- ; and the foreign force subducted as cA, and therefore the remain- A .A. ^ c^4 ing force as — -^ ; then (by the third Example) b will be equal to 1. m equal to 1, and n equal to 4 ; and therefore the angle of revolution be 1 — c •tween the apsides is equal to 180 <*/- — deg. Suppose that foreign force to be 357.45 parts less than the other force with which the body revolves in the ellipsis : that is, c to be -3 }y£ j ; A or T being equal to 1 ; and then l8(Vl~4c will be 18<Vfff Jf or 180.7623, that is, 180 deg., 45 min., 44 sec. Therefore the body, parting from the upper apsis, will arrive at the lower apsis with an angular motion of 180 deg., 45 min., 44 sec , and this angular motion being repeated, will return to the upper apsis ; and therefore the upper apsis in each revolution will go forward 1 deg., 31 min., 28 sec. The apsis of the moon is about twice as swift So much for the motion of bodies in orbits whose planes pass through the centre of force. It now remains to determine those motions in eccen trical planes. For those authors who treat of the motion of heavy bodies used to consider the ascent and descent of such bodies, not only in a per pendicular direction, but at all degrees of obliquity upon any given planes ; and for the same reason we are to consider in this place the motions of bodies tending to centres by means of any forces whatsoever, when those bodies move in eccentrical planes. These planes are supposed to be perfectly smooth and polished, so as not to retard the motion of the bodies in the least. Moreover, in these demonstrations, instead of the planes upon which those bodies roll or slide, and which are therefore tangent planes to the bodies, I shall use planes parallel to them, in which the centres of the bodies move, and by that motion describe orbits. And by the same method I afterwards determine the motions of bodies performer1 in curve superficies. SECTION X. Of the motion of bodies in given superficies, and of the reciprocal motion of fnnependulous bodies. PROPOSITION XL VI. PROBLEM XXXII. Any kind of centripetal force being supposed, and the centre of 'force •, atfft any plane whatsoever in which the body revolves, being given, and tint quadratures of curvilinear figures being allowed; it is required to de termine the motion of a body going off from a given place., with a given velocity, in the direction of a given right line in, that plane. SEC. X.J OF NATURAL PHILOSOPHY- 183 Let S be the centre of force, SC the least distance of that centre from the given plane, P a body issuing from the place P in the direction of the right line PZ, Q, the same body revolving in its trajectory, and PQ,R the trajectory itself which is required to be found, described in that given plane. Join CQ, Q.S, and if in Q,S we take SV proportional to the centripetal force with which the body is attracted to wards the centre S, and draw VT parallel to CQ, and meeting SC in T ; then will the force SV be resolved into two (by Cor. 2, of the Laws of Motion), the force ST, and the force TV ; of which ST aMracting the body in the direction of a line perpendicular to that plane, does not at all change its motion in that plane. But the action c f the other force TV, coinciding with the position of the plane itself, at tracts the body directly towards the given point C in that plane ; a«d t icreftre causes the body to move in this plane in the same manner as if the force S F were taken away, and the body were to revolve in free space about the centre C by means of the force TV alone. But there being given the centripetal force TV with which the body Q, revolves in free space about the given centre C, there is given (by Prop. XLII) the trajectory PQ.R which the body describes ; the place Q,, in which the body will be found at any given time ; and, lastly, the velocity of the body in that place Q,. And so e contra. Q..E.I. PROPOSITION XLV1L THEOREM XV. Supposing the centripetal force to be proportional to t/ie distance of the body from the centre ; all bodies revolving in any planes whatsoever will describe ellipses, and complete their revolutions in equal times ; and those which move in right lines, running backwards and forwards alternately ', will complete ttieir several periods of going and returning in the same times. For letting all things stand as in the foregoing Proposition, the force SV, with which the body Q, revolving in any plane PQ,R is attracted to wards the centre S, is as the distance SO. ; and therefore because SV and SQ,, TV and CQ, are proportional, the force TV with which the body is attracted towards the given point C in the plane of the orbit is as the dis tance CQ,. Therefore the forces with which bodies found in the plane PQ,R are attracted towaitis the point O, are in proportion to the distances equal to the forces with which the same bodies are attract-ed every way to wards the centre S ; and therefore the bodies will move in the same times, and in the same figures, in any plane PQR about the point C. n* they THE MATHEMATICAL PRINCIPLES [BOOK I. would do in free spaces about the centre S ; and therefore (by Cor. 2, Prop. X, ai d Cor. 2, Prop. XXXVIII.) they will in equal times either describe ellipses m that plane about the centre C, or move to and fro in right lines passing through the centre C in that plane; completing the same periods of time in all cases. Q.E.D. SCHOLIUM. The ascent and descent of bodies in curve superficies has a near relation to these motions we have been speaking of. Imagine curve lines to be de scribed on any plane, and to revolve about any given axes passing through the centre of force, and by that revolution to describe curve superficies ; and that the bodies move in such sort that their centres may be always found m those superficies. If those bodies reciprocate to and fro with an oblique ascent and descent, their motions will be performed in planes passing through tiie axis, and therefore in the curve lines, by whose revolution those curve superficies were generated. In those cases, therefore, it will be sufficient to consider thp motion in those curve lines. PROPOSITION XL VIII. THEOREM XVI. If a wheel stands npon the outside of a globe at right angles thereto, and revolving about its own axis goes forward in a great circle, the length of lite curvilinear path which any point, given in the perimeter of the wheel, hath described, since, the time that it touched the globe (which curvilinear path w~e may call the cycloid, or epicycloid), will be to double the versed sine of half the arc which since that time has touched the globe in passing over it, as the sn,m of the diameters of the globe and the wheel to the semi-diameter of the globe. PROPOSITION XLIX. THEOREM XVII. ff a wheel stand upon the inside of a concave globe at right angles there to, and revolving about its own axis go forward in one of the great circles of the globe, the length of the curvilinear path which any point, given in the perimeter of the wheel^ hath described since it toncJied the globe, imll be to the double of the versed sine of half the arc which in all that time has touched the globe in passing over it, as the difference of the diameters of the globe and the wheel to the semi-diameter of the globe. Let ABL be the globe. C its centre, BPV the wheel insisting thereon, E the centre of the wheel, B the point of contact, and P the given point in the perimeter of the wheel. Imagine this wheel to proceed in the great circle ABL from A through B towards L, and in its progress to revolve in such a manner that the arcs AB, PB may be always equal one to the other, :if;d the given point P in the peri meter of the wheel may describe in thf SEC. X.I OF NATURAL PHILOSOPHY. s 185 H mean time the curvilinear path AP. Let AP be the whole curvilinear path described since the wheel touched the globe in A, and the length cf this path AP will be to twice the versed sine of the arc |PB as 20 E to CB. For let the right line CE (produced if need be) meet the wheel in V, and join CP, BP, EP, VP ; produce CP, and let fall thereon the perpen dicular VF. Let PH, VH, meeting in H, touch the circle in P and V, and let PH cut YF in G, and to VP let fall the perpendiculars GI, HK. From the centre C with any interval let there be described the circle wow, cutting the right line CP in nt the perimeter of the wheel BP in o, and the curvilinear path AP in m ; and from the centre V with the interval Vo let there be described a circle cutting VP produced in q. Because the wheel in its progress always revolves about the point of con tact B. it is manifest that the right line BP is perpendicular to that curve line AP which the point P of the wheel describes, and therefore that the right line VP will touch this curve in the point P. Let the radius of the circle nmn be gradually increased or diminished so that at last it become equal to the distance CP ; and by reason of the similitude of the evanescent figure Pnn-mq, and the figure PFGVI, the ultimate ratio of the evanescent lined ae Pra, P//, Po, P<y, that is, the ratio of the momentary mutations of the curve AP, the right line CP, the circular arc BP, and the right line VP, will ••< iSS THE MATHEMATICAL PRINCIPLES [BOOK 1. the same as of the lines PV, PF, PG, PI, respectively. But since VF is perpendicular to OF, and VH to CV, and therefore the angles HVG, VCF equal: and the angle VHG (because the angles of the quadrilateral figure HVEP are right in V and P) is equal to the angle CEP, the triangles V HG, CEP will be similar ; and thence it will come to pass that as EP is to CE so is HG to HV or HP, and so KI to KP, and by composition or division as CB to CE so is PI to PK, and doubling the consequents asCB to 2CE so PI to PV, and so is Pq to Pm. Therefore the decrement of the line VP, that is, the increment of the line BY— VP to the increment of the curve line AP is in a given ratio of CB to 2CE, and therefore (by Cor. Lena. IV) the lengths BY— YP and AP, generated by those increments, arc in the same ratio. But if BY be radius, YP is the cosine of the angle BYP or -*BEP, and therefore BY— YP is the versed sine of the same angle, and therefore in this wheel, whose radius is ^BV, BY— YP will be double the versed sine of the arc ^BP. Therefore AP is to double the versed sine oi the arc ^BP as 2CE to CB. Q.E.D. The line AP in the former of these Propositions we shall name the cy cloid without the globe, the other in the latter Proposition the cycloid within the globe, for distinction sake. COR. 1. Hence if there be described the entire cycloid ASL, and the same be bisected in S, the lencth of the part PS will be to the length PV (which is the double of the sine of the angle YBP, when EB is radius) as 2CE to CB, and therefore in a given ratio. COR. 2. And the length of the semi-perimeter of the cycloid AS will be equal to a right line which is to the dumeter of the wheel BY as 2CF toCB. PROPOSITION L. PROBLEM XXXIII. To cause a pendulous body to oscillate in a given cycloid. Let there be given within the globe QYS de- scribed with the centre C, the cycloid QRS, bi sected in R, and meeting the superficies of the globe with its extreme points Q and S on either hand. Let there be drawn CR birxcting the arc QS in O, and let it be produced to A in such sort that CA may be to CO as CO to CR. About the centre C, with the interval CA, let there be described an exterior globe DAF ; and within this globe, by a wheel whose diameter is AO, let there be described two semi-cycloids AQ,, AS, touching the interior globe in Q, and S, and meeting the exterior globe in A. From that point A, with a thread APT in length equal to the line AR, let the body T depend, and oscillate in such manner between the two SlCC. X.J OF NATURAL PHILOSOPHY. 187 semi-cycloids AQ, AS, that, as often as the pendulum parts from the per pendicular AR, the upper part of the thread AP may be applied to that semi-cycloid APS towards which the motion tends, and fold itself round that curve line, as if it were some solid obstacle, the remaining part of the same thread PT which has not yet touched the semi-cycloid continuing straight. Then will the weight T oscillate in the given cycloid QRS. Q.E.F. For let the thread PT meet the cycloid QRS in T, and the circle QOS m V, and let 0V be drawn j and to the rectilinear part of the thread PT from the extreme points P and T let there be erected the perpendiculars BP, T W, meeting the right line CV in B and W. It is evident, from the construction and generation of the similar figures AS, SR, that those per pendiculars PB, TVV, cut off from CV the lengths VB, VVV equal the diameters of the wheels OA, OR. Therefore TP is to VP (which is dou ble the sine of the angle VBP when ^BV is radius) as BYV to BV, or AO -f-OR to AO, that is (since CA and CO, CO and CR; and by division AO and OR are proportional), as CA + CO to CA, or, if BV be bisected in E, as 2CE to CB. Therefore (by Cor. 1, Prop. XLIX), the length of the rectilinear part of the thread PT is always equal to the arc of the cycloid PS, and the whole thread APT is always equal to the half of the cycloid APS, that is (by Cor. 2, Prop. XLIX), to the length AR. And there fore contrariwise, if the string remain always equal to the length AR, the point T will always move in the given cycloid QRS. Q.E.D. COR. The string AR is equal to the semi-cycloid AS, and therefore has the same ratio to AC the semi-diameter of the exterior globe as the like semi-cycloid SR has to CO the semi-diameter of the interior globe. PROPOSITION LI. THEOREM XVIII. If a centripetal force tending on all sides to the centre C of a globe, be in all places as the distance of the place from the centre, and by this force alone acting upon it, the body T oscillate (in the manner above de scribed] in the perimeter of the cycloid QRS ; / say, that all the oscil lations, how unequal soever in tfiemselves, will be performed in equal times. For upon the tangent T W infinitely produced let fall the perpendicular CX, and join CT. Because the centripetal force with which the body T is impelled towards C is as the distance CT, let this (by Cor. 2, of the I ,aws) be resolved into the parts CX, TX, of which CX impelling the body directly from P stretches the thread PT, and by the resistance the 'rhread makes to it is totally employed, producing no other effect ; but the 3ther part TX, impelling the body transversely or towards X, directly accelerates the motion in the cycloid. Then it is plain that the accelera tion of the body, proportional to this accelerating force, will bo every 188 THE MATHEMATICAL PRINCIPLES [BOOK 1 moment as the length TX, that is (because CV\ WV, and TX, TW proportional to them are given), as the length TW, that is (by Cor. 1, Prop. XLIX) as the length of the arc of the cycloid TR. If there fore two pendulums APT, Apt, be unequally drawn aside from the perpendicular AR, and let fall together, their accelerations will be always as the arcs to be de scribed TR, tR. But the parts described at the beginning of the motion are as the accelerations, thai is, as the wholes that are to be described at the be ginning, and therefore the parts which remain to be described, and the subsequent accelerations proportional to those parts, are also as the wholes, and so on. Therefore the accelerations, and consequently the velocities generated, and the parts described with those velocities, and the parts to be described, are always as the wholes ; and therefore the parts to be described preserving a given ratio to each other will vanish together, that is, the two bodies oscillating will arrive together at the perpendicular AR. And since on the other hand the ascent of the pendulums from the lowest place R through the same cycloidal arcs with a retrograde motion, is retarded in the several places they pass through by the same forces by which their de scent was accelerated : it is plain that the velocities of their ascent and de scent through the same arcs are equal, and consequently performed in equal times ; and, therefore, since the two parts of the cycloid RS and RQ lying on either side of the perpendicular are similar and equal, the two pendu lums will perform as well the wholes as the halves of their oscillations in the same times. Q.E.D. ' COR. The force with which the body T is accelerated or retarded in any place T of the cycloid, is to the whole weight of the same body in the highest place S or Q, as the arc of the cycloid TR is to the arc SR or QR PROPOSITION LIL PROBLEM XXXIV. To define the velocities of the pendulums in the several places, and the times in which both the entire oscillations, and the several parts of them are performed. About any centre G, with the interval GH equal to the arc of the cycloid RS, describe a semi-circle HKM bisected by the semi-diameter GK. And if a centripe tal force proportional to the distance of the places from the centre tend to the centre G, and it be in the peri meter HIK equal to the centripetal force in the perime ter of the globe Q,OS tending towards its centre, and at the same time that the pendulum T is let fall from the highest place S, a body, as L, is let fall from H to G ; then because th< SEC. X.J OF NATURAL PHILOSOPHY. 189 forces which act upon the bodies are equal at the be ginning, and always proportional to the spaces to be described TR, LG, and therefore if TR and LG are equal, are also equal in the places T and L, it is plain that those bodies describe at the beginning equal spaces M ST, HL, and therefore are still acted upon equally, and continue to describe equal spaces. Therefore by Prop. XXXVIII, the time in which the body describes the arc ST is to the time of one oscillation, as the arc HI the time in which the body H arrives at L, to the semi-periphery HKM, the time in which the body H will come to M. And the velocity of the pendulous body in the place T is to its velocity in the lowest place R, that is, the velocity of the body H in the place L to its velocity in the place G, or the momentary increment of the line HL to the momentary increment of the line HG (the arcs HI, HK increasing with an equable flux) as the ordinato LI to the radius GK. or as v/SR2 — Til2 to SR. Hence, since in unequal oscillations there are described in equal time arcs proportional to the en tire arcs of the oscillations, there are obtained from the times given, both the velocities and the arcs described in all the oscillations universally. Which was first required. Let now any pendulous bodies oscillate in different cycloids described within different globes, whose absolute forces are also different ; and if the absolute force of any globe Q.OS be called V, the accelerative force with which the pendulum is acted on in the circumference of this globe, when it begins to move directly towards its centre, will be as the distance of the pendulous body from that centre and the absolute force of the globe con- junctly, that is, as CO X V. Therefore the lineola HY, which is as this accelerated force CO X V, will be described in a given time : and if there be erected the perpendicular YZ meeting the circumference in Z, the nascent arc HZ will denote that given time. But that nascent arc HZ is in the subduplicate ratio of the rectangle GHY, and therefore as v/GH X CO X V Whence the time of an entire oscillation in the cycloid Q,RS (it being as the semi-periphery HKM, wrhich denotes that entire oscillation, directly : and as the arc HZ which in like manner denotes a given time inversely) will be as GH directly and v/GH X CO X V inversely ; that is, because GH and SR are equal, as Vnr, . ™ or (by Cor. Prop. L,) as X/-TTVT- — UU X V AO X V Therefore the oscillations in all globes and cycloids, performed with what absolute forces soever, are in a ratio compounded of the subduplicate ratio of the length of the string directly, and the subduplicate ratio of the distance between the point of suspension and the centre of the globe inversely, and the subduplicate ratio of the absolute force of the globe inversely also Q.E.I. t90 THE MATHEMATICAL PRINCIPLES [Bo^K 1. COR. 1. Hence also the times of oscillating, falling, and revolving bodies may be compared among themselves. For if the diameter of the wheel with which the cycloid is described within the globe is supposed equal to the semi-diameter of the globe, the cycloid will become a right line passing through the centre of the globe, and the oscillation will be changed into a descent and subsequent ascent in that right line. Whence there is given both the time of the descent from any place to the centre, and the time equal to it in which the body revolving uniformly about the centre of the globe at any distance describes an arc of a quadrant For this time (by Case 2) is to the time of half the oscillation in any cycloid QJR.S as 1 to AR V AC' COR. 2. Hence also follow what Sir Christopher Wren and M. Huygevs have discovered concerning the vulgar cycloid. For if the diameter of the globe be infinitely increased, its sphacrical superficies will be changed into a plane, and the centripetal force will act uniformly in the direction of lines perpendicular to that plane, and this cycloid of our's will become the same with the common cycloid. But in that case the length of the arc of the cycloid between that plane and the describing point will become equal to four times the versed sine of half the arc of the wheel between the same plane and the describing point, as was discovered by Sir Christopher Wren. And a pendulum between two such cycloids will oscillate in a similar and equal cycloid in equal times, as M. Huygens demonstrated. The descent of heavy bodies also in the time of one oscillation will be the same as M. Huygens exhibited. The propositions here demonstrated are adapted to the true constitution of the Earth, in so far as wheels moving in any of its great circles will de scribe, by the motions of nails fixed in their perimeters, cycloids without the globe ; and pendulums, in mines and deep caverns of the Earth, must oscil late in cycloids within the globe, that those oscillations may be performed in equal times. For gravity (as will be shewn in the third book) decreases in its progress from the superficies of the Earth ; upwards in a duplicate ratio of the distances from the centre of the Earth ; downwards in a sim ple ratio of the same. PROPOSITION LIII. PROBLEM XXXV. Granting the quadratures of curvilinear figures, it is required to find the forces with which bodies moving in given curve lines may always perform their oscillations in equal times. Let the body T oscillate in any curve line STRQ,, whose axis is AR passing through the centre of force C. Draw TX touching that curve in any place of the body T, and in that tangent TX take TY equal to the arc TR. The length of that arc is known from the common methods used SEC. X. OF NATURAL PHILOSOPHY. 191 for the quadratures of figures. From the point Y draw the right line YZ perpendicular to the tangent. Draw CT meeting that perpendicular in Z, and the centripetal force will be proportional to the right line TZ. Q.E.I. For if the force with which the body is attracted from T towards C be expressed by the right line TZ taken proportional to it, that force will be resolved into two forces TY, YZ, of which YZ drawing the body in the direction of the length of the thread PT, docs not at all change its motion ; whereas the other force TY directly accelerates or retards its mction in the curve STRQ. Wherefore since that force is as the space to be described TR, the acceler ations or retardations of the body in describing two proportional parts (u greater arid a less) of two oscillations, will be always as those parts, and therefore will cause those parts to be described together. But bodies which continually describe together parts proportional to the wholes, will describe the wholes together also. Q,.E.l). COR. 1. Hence if the body T, hanging by a rectilinear thread AT from the centre A, describe the circular arc STRQ,, and in the mean time be acted on by any force tending downwards with parallel directions, which is to the uni form force of gravity as the arc TR to its sine TN, the times of the several oscillations will be equal. For because TZ, AR are parallel, the triangles ATN, ZTY are similar ; and there fore TZ will be to AT as TY to TN ; that is, if the uniform force of gravity be expressed by the given length AT, the force TZ. by which the oscillations become isochronous, will be to the force of gravity AT, as the arc TR equal to TY is to TN the sine of that arc. COR. 2. And therefore in clocks, if forces were impressed by some ma chine upon the pendulum which preserves the motion, and so compounded with the force of gravity that the whole force tending downwards should be always as a line produced by applying the rectangle under the arc TR and the radius AR to the sine TN, all the oscillations will become isochronous. PROPOSITION LIV. PROBLEM XXX YI. Granting the quadratures of curvilinear figures, it is required to find the times in which bodies by means of any centripetal force will descend or ascend in any curve lines described in, a plane passing through the centre of force. Let the body descend from any place S, and move in any curve ST/R given in a plane passing through the centre of force C. Join CS, and lei 192 THE MATHEMATICAL PRINCIPLES [BOOK 1 Q it be divided into innumerable equal parts, and let Dd be one of those parts. From the centre C, with the intervals CD, Cd, let the circles DT, dt be de scribed, meeting the curve line ST*R in T and t. And because the law of centripetal force is given. and also the altitude CS from which the body at first fell, there will be given the velocity of the body in any other altitude CT (by Prop. XXXIX). But the time in which the body describes the lineola Tt is as the length of that lineola, that is, as the secant of the angle /TC directly, and the velocity inversely. Lei, the ordinate DN, proportional to this time, be made perpendicular to the right line CS at the point D, and because Dd is given, the rectangle Dd X DN, that is, the area DNwc?, will be proportional to the same time. Therefore if PN/?, be a curve line in which the point N is perpetually found, and its asymptote be the right line SQ, standing upon the line CS at right angles, the area SQPJN D will be proportional to the time in which the body in its descent hath described the line ST ; and therefore that area being found, the time is also given. Q.E.I. PROPOSITION LV. THEOREM XIX. If a body move in any curve superficies, whose axis passes through the centre of force, and from the body a perpendicular be let fall iipon the axis \ and a line parallel and equal thereto be drawn from any given point of the axis ; I say, that this parallel line will describe an area proportional to the time. Let BKL be a curve superficies, T a body revolving in it, STR a trajectory which the body describes in the same, S the beginning of the trajectory, OMK the axis of the curve superficies, TN a right line let fall perpendic ularly from the body to the axis ; OP a line parallel and equal thereto drawn from the given point O in the axis ; AP the orthogra phic projection of the trajectory described by the point P in the plane AOP in which the revolving line OP is found : A the beginning of that projection, answering to the point S ; TO a right line drawn from the body to the centre ; TG a part thereof proportional to the centripetal force with which the body tends towards the centre C ; TM a right line perpendicular to the curve superficies ; TI a part thereof proportional to the force of pressure with which the body urges SEC. X.] OF NATURAL PHILOSOPHY. 193 the superficies, and therefore with which it is again repelled by the super ficies towards M ; PTF a right line parallel to the axis and passing through the body, and OF, IH right lines let fall perpendicularly from the points G and I upon that parallel PHTF. I say, now. that the area AGP, de scribed by the radius OP from the beginning of the motion, is proportional to the time. For the force TG (by Cor. 2, of the Laws of Motion) is re solved into the forces TF, FG ; and the force TI into the forces TH, HI ; but the forces TF, TH, acting in the direction of the line PF perpendicular to the plane AOP, introduce no change in the motion of the body but in a di rection perpendicular to that plane. Therefore its motion, so far as it has the same direction with the position of the plane, that is, the motion of the point P, by which the projection AP of the trajectory is described in that plane, is the same as if the forces TF, TH were taken away, and the body wei e acted on by the forces FG, HI alone ; that is, the same as ,f the body were to describe in the plane AOP the curve AP by means of a centripetal force tending to the centre O, and equal to the sum of the forces FG and HI. But with such a force as that (by Prop. 1) the area AOP will be de scribed proportional to the time. Q.E.D. COR. By the same reasoning, if a body, acted on by forces tending to two or more centres in any the same right line CO, should describe in a free space any curve line ST, the area AOP would be always proportional to the time. PROPOSITION LVI. PROBLEM XXXVII. Granting the quadratures of curvilinear figures, and supposing that there are given both the law of centripetal force tending to a given cen tre, and the curve superficies whose axis passes through that centre ; it is required to find the trajectory which a body will describe in that superficies, when going off from a given place with a given velocity, and in a given direction in that superficies. The last construction remaining, let the body T go from the given place S, in the di rection of a line given by position, and turn into the trajectory sought STR, whose ortho graphic projection in the plane BDO is AP. And from the given velocity of the body in the altitude SC, its velocity in any other al titude TC will be also given. With that velocity, in a given moment of time, let the body describe the particle Tt of its trajectory, and let P/? be the projection of that particle described in the plane AOP. Join Op, and a little circle being described upon the curve superficies about the centre T 13 194 THE MATHEMATICAL PRINCIPLES [BOOK I with the interval TV let the projection of that little circle in the plane AOP be the ellipsis pQ. And because the magnitude of that little circle T/, and TN or PO its distance from the axis CO is also given, the ellipsis pQ, will be given both in kind and magnitude, as also its position to the right line PO. And since the area PO/? is proportional to the time, and therefore given because the time is given, the angle POp will be given. And thence will be given jo the common intersection of the ellipsis and. the right line Op, together with the angle OPp, in which the projection APp of the tra jectory cuts the line OP. But from thence (by conferring Prop. XLI, with Us 2d Cor.) the mariner of determining the curve APp easily appears. Then from the several points P of that projection erecting to the plane AOP, the perpendiculars PT meeting the curve superficies in T, there will be £iven the several points T of the trajectory. Q.E.I. SECTION XL f 'f the motions of bodies tending' to each other with centripetal forces. I have hitherto been treating of the attractions of bodies towards an im movable centre; though very probably there is no such thing existent in nature. For attractions are made towards bodies, and the actions of the bodies attracted and attracting are always reciprocal and equal, by Law III ; BO that if there are two bodies, neither the attracted nor the attracting body is truly at rest, but both (by Cor. 4, of the Laws of Motion), being as it were mutually attracted, revolve about a common centre of gravity. And if there be more bodies, which are either attracted by one single one which is attracted by them again, or which all of them, attract each other mutu ally , these bodies will be so moved among themselves, as that their common centre of gravity will either be at rest, or move uniformly forward in a right line. I shall therefore at present go on to treat of the motion of bodies mutually attracting each other ; considering the centripetal forces as attractions ; though perhaps in a physical strictness they may more truly be called impulses. But these propositions are to be considered as purely mathematical; and therefore, laying aside all physical considerations, I make use of a familiar way of speaking, to make myself the more easily understood by a mathematical reader. PROPOSITION LVII. THEOREM XX. Two bodies attracting each other mutually describe similar figures about their common centre of gravity, and about each other mutually. For the distances of the bodies from their common centre of gravity are leciprocally as the bodies; and therefore in a given ratio to each other: *nd thence, bv composition of ratios, in a given ratio to the whole distance SEC. XL] OF NATURAL PHILOSOPHY. 195 between the bodies. Now these distances revolve about their common term with an equable angular motion, because lying in the same right line they never change their inclination to each other mutually But right lines that are in a given ratio to each other, and revolve about their terms with an equal angular motion, describe upon planes, which either rest with those terms, or move with any motion not angular, figures entirely similar round those terms. Therefore the figures described by the revolution ot these distances are similar. Q.E.D. PROPOSITION LVIll. THEOREM XXI. If two bodies attract each other mutually with forces of any kind, and in the mean time revolve about the common centre of gravity ; I say, that, by the same forces, there may be described round either body un moved ajigure similar and equal to the figures ivhich the bodies so moving describe round each other mutually. Let the bodies S and P revolve about their common centre of gravity C, proceeding from S to T, and from P to Q,. From the given point s lot there be continually drawn sp, sq, equal and parallel to SP, TQ, ; and the ;urve pqv, which the point p describes in its revolution round the immovable point s, will be similar and equal to the curves which the bodies S and P' describe about each other mutually ; and therefore, by Theor. XX, similar to the curves ST and PQ,V which the same bodies describe about their common centre of gravity C • and that because the proportions of the lines SC. CP, and SP or sp, to each other, are given. CASE 1. The common centre of gravity C (by Cor. 4, of the Laws of Mo tion) is either at rest, or moves uniformly in a right line. Let us first suppose it at rest, and in s and p let there be placed two bodies, one im movable in s, the other movable in p, similar and equal to the bodies S arid P. Then let the right lines PR and pr touch the curves PQ, and pq ki P and p, and produce CQ, and sq to R and r. And because the figures CPRQ, sprq are similar, RQ, will be to rq as CP to sp, and therefore in a given ratio. Hence if the force with which the body P is attracted to wards the body S, and by consequence towards the intermediate point the centre C, were to the force with which the body p is attracted towards the Centre 5. in the same given ratio, these forces would in equal times attract 196 THE MATHEMATICAL PRINCIPLES |BoOK 1 the bodies from the tangents PR, pr to the arcs PQ, pq, through the in tervals proportional to them RQ,, rq ; and therefore this last force (tending to s) would make the body p revolve in the curve pqv, which would becomr similar to the curve PQV, in which the first force obliges the body P i( revolve ; and their revolutions would be completed in the same timeg But because those forces are not to each other in the ratio of CP to sp, bu; (by reason of the similarity and equality of the bodies S and s, P and / and the equality of the distances SP, sp) mutually equal, the bodies ii equal times will be equally drawn from the tangents; and therefore tLV the body p may be attracted through the greater interval rq, there is re quired a greater time, which will be in the subduplicate ratio of the inter vals ; because, by Lemma X, the spaces described at the very beginning ol the motion are in a duplicate ratio of the times. Suppose, then the velocity of the body p to be to the velocity of the body P in a subduplicate ratio of the distance sp to the distance CP, so that the arcs pq, PQ, which are in a simple proportion to each other, may be described in times that are in n subduplicate ratio of the distances ; and the bodies P, p, always attracted by equal forces, will describe round the quiescent centres C and s similar figures PQV, pqv, the latter of which pqv is similar and equal to the figure ivhich the body P describes round the movable body S. Q.E.I) CASE 2. Suppose now that the common centre of gravity, together with the space in which the bodies are moved among themselves, proceeds uni formly in a right line ; and (by Cor. 6, of the Laws of Motion) all the mo tions in this space will be performed in the same manner as before ; and therefore the bodies will describe mutually about each other the same fig ures as before, which will be therefore similar and equal to the figure pqv. Q.E.D. COR. 1. Hence two bodies attracting each other with forces proportional to their distance, describe (by Prop. X) both round their common centre ol gravity, and round each other mutually concentrical ellipses ; and, vice versa, if such figures are described, the forces are proportional to the dis tances. COR. 2. And two bodies, whose forces are reciprocally proportional to the square of their distance, describe (by Prop. XI, XII, XIII), both round their common centre of gravity, and round each other mutually, conic sec tions having their focus in the centre about which the figures are described. And, vice versa, if such figures are described, the centripetal forces are re ciprocally proportional to the squares of the distance. COR. 3. Any two bodies revolving round their common centre of gravity describe areas proportional to the times, by radii drawn both to that centre and to each other mutually- >EC. XL] OF NATURAL PHILOSOPHY. 197 PROPOSITION LIX. THEOREM XXII. The periodic time of two bodies S and P revolving round their common centre of gravity C,is to the periodic time of one of the bwlies 1? re volving round the other S remaining unmoved, and describing a fig ure similar and equal to those which the bodies describe about each other mutually r, in a subduplicate ratio of the other body S to the sii/rn of the bodies S -f P. For, by the demonstration of the last Proposition, the times in which any similar arcs PQ and pq are described are in a subduplicate ratio of the distances CP and SP, or sp, that is, in a subduplicate ratio of the ody S to the sum of the bodies S + P. And by composition of ratios, the sums of the times in which all the similar arcs PQ and pq are described, that is, the whole times in which the whole similar figures are described are in the same subduplicate ratio. Q.E.D. PROPOSITION LX. THEOREM XXIII. If two bodies S and P, attracting each other with forces reciprocally pro portional to the squares of their distance, revolve about their common centre of gravity ; I say, that the principal axis of the ellipsis which either of the bodies, as P, describes by this motion about the other S, will be to the principal axis of the ellipsis, which the same body P may describe in the same periodical time about the other body S quiescent, as the sum of the two bodies S + P to the first of two mean, propor tionals between that sum and the other body S. For if the ellipses described were equal to each other, their periodic times by the last Theorem would be in a subduplicate ratio of the body S to the sum of the bodies S 4- P. Let the periodic time in the latter ellipsis be diminished in that ratio, and the periodic times will become equal ; but, by Prop. XV, the principal axis of the ellipsis will be diminished in a ratio sesquiplicate to the former ratio ; that is, in a ratio to which the ratio of S to S 4- P is triplicate ; and therefore that axis will be to the principal axis of the other ellipsis as the first of two mean proportionals between S -f- P and S to S 4- P. And inversely the principal axis of the ellipsis de scribed about the movable body will be to the principal axis of that described round the immovable as S + P to the first of two mean proportionals be tween S 4- P and S. Q.E.D. PROPOSITION LXI. THEOREM XXIV. If two bodies attracting each other with any kind of forces, and not otherwise agitated or obstructed, are moved in any manner whatsoever, those 'motions will be the same as if they did not at all attract each other mutually, but were both attracted with the same forces by a third body placed in their common centre of gravity ; and the law of the 198 THE MATHEMATICAL PRINCIPLES [BOOK I attracting Jones will be the sam# in respect of the distance of the. bodies from, the common centre, as in respect of the distance between the two bodies. For those forces with which the bodies attract each other mutually, by tending to the bodies, tend also to the common centre of gravity lying di rectly between them ; and therefore are the same as if they proceeded from 'an intermediate body. QJG.D. And because there is given the ratio of the distance of either body from that common centre to the distance between the two bodies, there is given, of course, the ratio of any power of one distance to the same power of the . ther distance ; and also the ratio of any quantity derived in any manner from one of the distances compounded any how with given quantities, to another quantity derived in like manner from the other distance, and as many given quantities having that given ratio of the distances to the first Therefore if the force with which one body is attracted by another be di rectly or inversely as the distance of the bodies from each other, or as any power of that distance ; or, lastly, as any quantity derived after any man ner from that distance compounded with given q-uantities ; then will the same force with which the same body is attracted to the common centre of gravity be in like manner directly or inversely as the distance of the at tracted body from the common centre, or as any power of that distance ; or, lastly, as a quantity derived in like sort from that distance compounded with analogous given quantities. That is, the law of attracting force will be the same with respect to both distances. Q,.E.D. PROPOSITION LXII. PROBLEM XXXVIII. To determine the motions of two bodies which attract each other with forces reciprocally proportional to the squares of the distance between them, aflid are, let fall from given places. The bodies, by the last Theorem, will be moved in the same manner as if they were attracted by a third placed in the common centre of their gravity ; and by the hypothesis that centre will be quiescent at the begin ning of their motion, and therefore (by Cor. 4, of the Laws of Motion) will be always quiescent. The motions of the bodies are therefore to be deter mined (by Prob. XXV) in the same manner as if they were impelled by forces tending to that centre: and then we shall have the motions of the bodies attracting each other mutually. Q.E.I. PROPOSITION LXIII. PROBLEM XXXIX. To determine the motions of two bodies attracting each other with forces reciprocally proportional to the squares of their distance, and going off from given places in, given directions with given velocities. The motions of the bodies at the beginning being given, there is given SEC. XL] OF NATURAL PHILOSOPHY. 1 % also the uniform motion of the common centre of gravity, and the motion of the space which moves along with this centre uniformly in a right line, and also the very first, or beginning motions of the bodies in respect of this space. Then (by Cor. 5, of the Laws, and the last Theorem) the subse quent motions will be performed in the same manner in that space, as if that space together with the common centre of gravity were at rest, and as if the bodies did not attract each other, but were attracted by a third body placed in that centre. The motion therefore in this movable space of each body going off from a given place, in a given direction, with a given velo city, and acted upon by a centripetal force tending to that centre, is to be determined by Prob. IX and XXVI, and at the same time will be obtained the motion of the other round the same centre. With this motion com pound the uniform progressive motion of the entire system of the space and the bodies revolving in it, and there will be obtained the absolute motion of the bodies in immovable space. Q..E.I. PROPOSITION LXIV. PROBLEM XL. Supposing forces with which bodies mutually attract each other to in crease in a simple ratio of their distances from the centres ; it is ro- quired to find the motions of several bodies among themselves. Suppose the first two bodies T and L to have their common centre of gravity in L). These, by Cor. 1, Theor. XXI, will S describe ellipses having their centres in D, the magnitudes of which ellipses are known by Prob. V. J- -- \- ? L Let now a third body S attract the two former T and L with the accelerative forces ST, SL, and let it be attract ed again by them. The force ST (by Cor. 2, of the Laws of Motion) is resolved into the forces SD, DT ; and the force SL into the forces SD and DL. Now the forces DT, DL. which are as their sum TL, and therefore as the accelerative forces with which the bodies T and L attract each other mutually, added to the forces of the bodies T and L, the first to the first, and the last to the last, compose forces proportional to the distances DT and DL as before, but only greater than those former forces : and there fore (by Cor. 1, Prop. X, and Cor. l,and 8, Prop. IV) they will cause those bodies to describe ellipses as before, but with a swifter motion. The re maining accelerative forces SD and DL, by the motive forces SD X Tand SD X L, which are as the bodies attracting those bodies equally and in the direction of the lines TI, LK parallel to DS, do not at all change their situ ations with respect to one another, but cause them equally to approach to the line IK ; which must be imagined drawn through the middle of the body S, and perpendicular to the line DS. But that approach to the line 200 THE MATHEMATICAL PRINCIPLES [BoOK I. IK will be hindered by causing the system of the bodies T and L on one side, and the body S on the other, with proper velocities, to revolve round the common centre of gravity C. With such a motion the body S, because the sum of the motive forces SD X T and SD X L is proportional to the distance OS, tends to the centre C, will describe an ellipsis round the same centre C; and the point D, because the lines CS and CD are proportional, will describe a like ellipsis over against it. But the bodies T and L, at tracted by the motive forces SD X T and SD X L, the first by the first, and the last by the last, equally and in the direction of the parallel lines TI and LK, as was said before, will (by Cor. 5 and 6, of the Laws of Motion) continue to describe their ellipses round the movable centre D, as before. Q.E.I. Let there be added a fourth body V, and, by the like reasoning, it will be demonstrated that this body and the point C will describe ellipses about the common centre of gravity B ; the motions of the bodies T, L, and S round the centres D and C remaining the same as before ; but accelerated. Arid by the same method one may add yet more bodies at pleasure. Q..E.I. ^This would be the case, though the bodies T and L attract each other mutually with accelerative forces either greater or less than those with which they attract the other bodies in proportion to their distance. Let all the mutual accelerative attractions be to each other as the distances multiplied into the attracting bodies ; and from what has gone before it will easily be concluded that all the bodies will describe different ellipses with equal periodical times about their common centre of gravity B, in an immovable plane. Q.E.I. PROPOSITION LXV. THEOREM XXV. Bodies, whose forces decrease in a duplicate ratio of their distances from their centres, 'may move among" themselves in ellipses ; and by radii drawn to the foci may describe areas proportional to the times very nearly. In the last Proposition we demonstrated that case in which the motions will be performed exactly in ellipses. The more distant the law of the forces is from the law in that case, the more will the bodies disturb each other's motions ; neither is it possible that bodies attracting each other mutually according to the law supposed in this Proposition should move exactly in ellipses, unless by keepirg a certain proportion of distances from each other. However, in the following crises the orbits will not much dif fer from ellipses. CASE I. Imagine several lesser bodies to revolve about some very great one at different distances from it, and suppose absolute forces tending to rvery one of the bodies proportional to each. And because (by Cor. 4, ol the I aws) the common centre of gravity of them all is either at rest, 01 iSEC. XL] OF NATURAL PHILOSOPHY. 20 i moves uniformly forward in a right line, suppose the lesser bodies so small that the groat body may be never at a sensible distance from that centre ; and then the great body will, without any sensible error, be either at rest, or move uniformly forward in a right line; and the lesser will revolve about that great one in ellipses, and by radii drawn thereto will describe areas proportional to the times ; if we except the errors that may be intro duced by the receding of the great body from the common centre of gravity, or by the mutual actions of the lesser bodies upon each other. But the lesser bodies may be so far diminished, as that this recess and the mutual actions of the bodies on each other may become less than any assignable; and therefore so as that the orbits may become ellipses, and the areas an swer to the times, without any error that is not less than any assignable. Q.E.O. CASE 2. Let us imagine a system of lesser bodies revolving about a very great one in the manner just described, or any other system of two bodies revolving about each other to be moving uniformly forward in a right line, and in the mean time to be impelled sideways by the force of another vastly greater body situate at a great distance. And because the equal accelerative forces with which the bodies are impelled in parallel directions do not change the situation of the bodies with respect to each other, but only oblige the whole system to change its place while the parts still retain their motions among themselves, it is manifest that no change in those motions of the attracted bodies can arise from their attractions towards the greater, unless by the inequality of the accelerative attractions, or by the inclinations of the lines towards each other, in whose directions the attractions are made. Suppose, therefore, all the accelerative attractions made towards the great body to be among themselves as the squares of the distances reciprocally ; and then, by increasing the distance of the great body till the differences of fhe right lines drawn from that to the others in respect of their length, and the inclinations of those lines to each other, be less than any given, the mo tions of the parts of the system will continue without errors that are not less than any given. And because, by the small distance of those parts from each other, the whole system is attracted as if it were but one body, it will therefore be moved by this attraction as if it were one body ; that is, its centre of gravity will describe about the great bod/ one of the conic sec tions (that is, a parabola or hyperbola when the attraction is but languid and an ellipsis when it is more vigorous) ; and by radii drawn thereto, it will describe areas proportional to the times, without any errors but thos« which arise from the distances of the parts, which are by the supposition exceedingly small, and may be diminished at pleasure. Q,.E.O. By a like reasoning one may proceed to more compounded cases in in- finitum. COR 1. In the second Case, the nearer the very great body approaches to ^0'^ THE MATHEMATICAL PRINCIPLES [CoOK I the system of two or more revolving bodies, the greater will the pertur bation be of the motions of the parts of the system among themselves; be cause the inclinations of the lines drawn from that great body to those parts become greater ; and the inequality of the proportion is also greater. COR. 2. But the perturbation will be greatest of all, if we suppose the uccelerative attractions of the parts of the system towards the greatest body of all are not to each other reciprocally as the squares of the distances from that great body ; especially if the inequality of this proportion be greater than the inequality of the proportion of the distances from the great body. For if the accelerative force, acting in parallel directions and equally, causes no perturbation in the motions of the parts of the system, it must of course, when it acts unequally, cause a perturbation some where, which will be greater or less as the inequality is greater or less. The excess of the greater impulses acting upon some bodies, and not acting upon others, must necessarily change their situation among themselves. And this perturbation, added to the perturbation arising from the inequality and inclination of the lines, makes the whole perturbation greater. COR. '*. Hence if the parts of this system move in ellipses or circles without any remarkable perturbation, it is manifest that, if they are at all impelled by accelerative forces tending to any other bodies, the impulse is very weak, or else is impressed very near equally and in parallel directions upon all of them. PROPOSITION LXVL THEOREM XXVI. Tf three bodies whose forces decrease in a duplicate ratio of the distances attract each other mutually ; and the accelerative attractions of any two towards the third be between themselves reciprocally as the squares, of the distances ; and the two least revolve about the greatest ; I say, that the interior of the tivo revolving bodies will, by radii drawn to the innermost and greatest, describe round thai body areas more propor tional to the times, and a figure more approaching to that of an ellip sis having its focus in the point of concourse of the radii, if that great body be agitated by those attractions, than it would do if lhat great body were not attracted at all by the lesser, but remained at rest ; or than it would if that great body were very much more or very much less attracted, <>r very much more or very much less agitated, by the attractions. This appears plainly enough from the demonstration of the second Corollary of tl.e foregoing Proposition; but it may be made out after this manner by a way of reasoning more distinct and more universally convincing. CASE 1. Let the lesser bodies P and S revolve in the same plane about the greatest body T, the body P describing the interior orbit PAB, and S SEC. XI.J OF NATURAL PHILOSOPHY. 203 the exterior orbit ESE. Let SK be the mean distance of the bodies P and S ; and let the accelerative attraction of the body P towards S, at that mean distance, be expressed by that line SK. Make SL to SK as the E C square of SK to the square of SP, and SL will be the accelerative attrac tion of the body P towards S at any distance SP. Join PT, and draw LM parallel to it meeting ST in M; and the attraction SL will be resolv ed (by Cor. 2. of the Laws of Motion) into the attractions SM, LM. And so the body P will be urged with a threefold accelerative force. One of these forces tends towards T, and arises from the mutual attraction of the bodies T and P. By this force alone the body P would describe round the body T, by the radius PT, areas proportional to the times, and an ellipsis whose focus is in the centre of the body T ; and this it would do whether the body T remained unmoved, or whether it were agitated by that attraction. This appears from Prop. XI, and Cor. 2 and 3 of Theor. XXI. The other force is that of the attraction LM, which, because it tends from P to T, will be superadded to and coincide with the former force ; and cause the areas to be still proportional to the times, by Cor. 3, Theor. XXI. But because it is not reciprocally proportional to the square of the distance PT, it will compose, when added to the former, a force varying from that proportion : which variation will be the greater by how much the proportion of this force to the former is greater, cceteris paribus. Therefore, since by Prop. XI, and by Cor. 2, Theor. XXI, the force with which the ellipsis is described about the focus T ought to be directed to that focus, and to be reciprocally proportional to the square of the distance PT, that compounded force varying from that proportion will make the orbit PAB vary from the figure of an ellipsis that has its focus in the point 'I1 ; and so much the more by how much the variation from that proportion is greater ; and by consequence by how much the proportion of the second force LM to the first force is greater, cceteris paribus. But now the third force SM, attracting the body P in a direction parallel to ST, composes with the other forces a new force which is no longer directed from P to T : and which varies so much more from this direction by how much the proportion of this third force to the other forces is greater, cceteris paribus ; arid therefore causes the body P to describe, by the radius TP, areas no longer proportional to the times : and therefore makes the variation from that proportionality so much greater by how much the proportion of this force to the others is greater. But this third force will increase the variation of the orbit PAB from th* THE MATHEMATICAL PRINCIPLES [BOOK 1 elliptical figure before-mentioned upon two accounts ; first because that force is not directed from P to T ; and, secondly, because it is not recipro cally proportional to the square of the distance PT. These things being premised, it is manifest that the areas are then most nearly proportional to the times, when that third force is the least possible, the rest preserving their former quantity ; and that the orbit PAB does then approach nearest to the elliptical figure above-mentioned, when both the second and third, but especially the third force, is the least possible; the first force remain ing in its former quantity. Let the accelerative attraction of the body T towards S be expressed by the line SN ; then if the accelerative attractions SM and SN were equal, these, attracting the bodies T and P equally and in parallel directions would not at all change their situation with respect to each other. The mo tions of the bodies between themselves would be the same in that case as if those attractions did not act at all, by Cor. 6, of the Laws of Motion. And, by a like reasoning, if the attraction SN is less than the attraction SM, it will take away out of the attraction SM the part SN, so that there will re main only the part (of the attraction) MN to disturb the proportionality of the areas and times, and the elliptical figure of the orbit. And in like manner if the attraction SN be greater than the attraction SM, the pertur bation of the orbit and proportion will be produced by the difference MN alone. After this manner the attraction SN reduces always the attraction SM to the attraction MN, the first and second attractions rema ning per fectly unchanged ; and therefore the areas and times come then nearest to proportionality, and the orbit PAB to the above-mentioned elliptical figure, when the attraction MN is either none, or the least that is possible; that is, when the accelerative attractions of the bodies P and T approach as near as possible to equality ; that is, when the attraction SN is neither none at all, nor less than the least of all the attractions SM, but is, as it were, a mean between the greatest and least of all those attractions SM, that is, not much greater nor much less than the attraction SK. Q.E.D. CASE 2. Let now the lesser bodies P. S, revolve about a greater T in dif ferent planes ; and the force LM, acting in the direction of the line PT situate in the plane of the orbit PAB, will have the same effect as before ; neither will it draw the body P from the plane of its orbit. But the other force NM acting in the direction of a line parallel to ST (and which, there fore, when the body S is without the line of the nodes is inclined to the plane of the orbit PAB), besides the perturbation of the motion just now spoken of as to longitude, introduces another perturbation also as to latitude, attracting the body P out of the plane of its orbit. And this perturbation, in any given situation of the bodies P and T to each other, will be as the generating force MN ; and therefore becomes least when the force MN is least, that is (as was just now shewn), where the attraction SN is not nrirb greater nor much less than the attraction SK. Q.E.D. SK-C. XL] OF NATURAL PHILOSOPHY. 205 COR. 1. Hence it may be easily collected, that if several less bodies P 8, R, &c.; revolve about a very great body T, the motion of the innermost revolving body P will be least disturbed by the attractions of the others. when the great body is as well attracted and agitated by the rest (accord ing to the ratio of the accelerative forces) as the rest are by each other mutually. COR. 2. In a system of three bodies, T, P, S, if the accelerative attrac tions of any two of them towards a third be to each other reciprocally as the squares of the distances, the body P, by the radius PT, will describe its area about the body T swifter near the conjunction A and the opposition B than it will near the quadratures C arid D. For every force with which the body P is acted on and the body T is not, and which does not act in the direction of the line PT, does either accelerate or retard the description of the area, according as it is directed, whether in consequentia or in cwtecedentia. Such is the force NM. This force in the passage of the body P frcm C to A is directed in consequentia to its motion, and therefore accelerates it; then as far as D in atttecedentia, and retards the motion; then in, con sequentia as far as B ; and lastly in antecedentia as it moves from B to C. COR. 3. And from the same reasoning it appears that the body P ccBteris paribuSj moves more swiftly in the conjunction and opposition than in the quadratures. COR. 4. The orbit of the body P, cc&teris paribus, is more curve at the quadratures than at the conjunction and opposition. For the swifter bodies move, the less they deflect from a rectilinear path. And besides the force KL, or NM, at the conjunction and opposition, is contrary to the force with which the body T attracts the body P, and therefore diminishes that force ; but the body P will deflect the less from a rectilinear path the less it is impelled towards the body T. COR. 5. Hence the body P, cceteris paribus, goes farther from the body T at the quadratures than at the conjunction and opposition. This is said, E C_ L B however, supposing no regard had to the motion of eccentricity. For if the orbit of the body P be eccentrical, its eccentricity (as will be shewn presently by Cor. 9) will be greatest when the apsides are in the syzy- gies; and thence it may sometimes come to pass that the body P. in its near approach to the farther apsis, may go farther from the body T at the syzygies than at the quadratures. COR. 6. Because the centripetal force of the central body T, by which 206 THE MATHEMATICAL PRINCIPLES [BOOK. 1 the body P is retained in its orbit, is increased at the quadratures by tho addition caused by the force LM, and diminished at the syzygies by the subduction caused by the force KL, and, because the force KL is greater than LM, it is more diminished than increased ; and, moreover, since that centripetal force (by Cor. 2, Prop. IV) is in a ratio compounded of the sim ple ratio of the radius TP directly, and the duplicate ratio of the periodi cal time inversely ; it is plain that this compounded ratio is diminished by the action of the force KL ; and therefore that the periodical time, supposing the radius of the orbit PT to remain the same, will be increased, and that in the subduplicate of that ratio in which the centripetal force is diminish ed ; and, therefore, supposing this radius increased or diminished, the peri odical time will be increased more or diminished less than in the sesquipli- cate ratio of this radius, by Cor. 6, Prop. IV. If that force of the central body should gradually decay, the body P being less and less attracted would go farther and farther from the centre T ; and, on the contrary, if it were increased, it would draw nearer to it. Therefore if the action of the distant body S, by which that force is diminished, were to increase and decrease by turns, the radius TP will be also increased and diminshed by turns ; and the periodical time will be increased and diminished in a ratio com pounded of the sesquiplicate ratio of the radius, and of the subduplicate oi that ratio in which the centripetal force of the central body T is dimin ished or increased, by the increase or decrease of the action of the distant body S. COR. 7. It also follows, from what was before laid down, that the axis of the ellipsis described by the body P, or the line of the apsides, does as to its angular motion go forwards and backwards by turns, but more for wards than backwards, and by the excess of its direct motion is in the whole carried forwards. For the force with which the body P is urged to the body T at the quadratures, where the force MN vanishes, is compound ed of the force LM and the centripetal force with which the body T at tracts the body P. The first force LM, if the distance PT be increased, is increased in nearly the same proportion with that distance, and the other force decreases in the duplicate ratio of the distance ; and therefore the sum of these two forces decreases in a less than the duplicate ratio of the distance PT ; and therefore, by Cor. 1, Prop. XLV, will make the line of the apsides, or, which is the same thing, the upper apsis, to go backward. But at the conjunction and opposition the force with which the body P is urged towards the body T is the difference of the force KL, and of the force with which the body T attracts the body P ; and that difference, be cause the force KL is very nearly increased in the ratio of the distance PT, decreases in more -than the duplicate ratio of the distance PT ; and therefore, by' Cor. 1, Prop. XLV, causes the line of the apsides to go for wards. In the places between the syzygies and the quadratures, the motion SEC. Xl.J OF NATURAL PHILOSOPHY. 207 of the line of the apsides depends upon both < f these causes conjuncdy, so that it either goes forwards or backwards in proportion to the excess ol one of these causes above the other. Therefore since the force KL in the syzygies is almost twice as great as the force LM in the quadratures, the excess will be on the side of the force KL, and by consequence the line of the apsides will be carried forwards. The truth of this arid the foregoing IE Corollary will be more easily understood by conceiving the system of the two bodies T and P to be surrounded on every side by several bodies S, S, S, dec., disposed about the orbit ESE. For by the actions of these bo dies the action of the body T will be diminished on every side, and decrease in more than a duplicate ratio of the distance. COR. 8. IJut since the progress or regress of the apsides depends upon the decrease of the centripetal force, that is, upon its being in a greater or less ratio than the duplicate ratio of the distance TP, in the passage of the body from the lower apsis to the upper ; and upon a like increase in its return to the lower apsis again ; and therefore becomes greatest where the proportion of the force at the upper apsis to the force at the lower ap sis recedes farthest from the duplicate ratio of the distances inversely ; it is plain, that, when the apsides are in the syzygies, they will, by reason of the subducting force KL or NM — LM, go forward more swiftly ; and in the quadratures by the additional force LM go backward more slowly. Because the velocity of the progress or slowness of the regress is continued for a long time ; this inequality becomes exceedingly great. COR. 9. If a body is obliged, by a force reciprocally proportional to the square of its distance from any centre, to revolve in an ellipsis round that centre ; and afterwards in its descent from the upper apsis to the lower apsis, that force by a perpetual accession of new force is increased in more than a duplicate ratio of the diminished distance ; it is manifest that the body, being impelled always towards the centre by the perpetual accession of this new force, will incline more towards that centre than if it were urged by that force alone which decreases in a duplicate ratio of the di minished distance, and therefore will describe an orbit interior to that elliptical orbit, and at the lower apsis approaching nearer to the centre than before. Therefore the orbit by the accession of this new force will become more eccentrical. If now, while the body is returning from the lower to the upper apsis, it should decrease by the same degrees by which it increases before the body would return to its first distance; and there- THE MATHEMATICAL PRINCIPLES [BOOK I. fore if the force decreases in a yet greater ratio, the body, being now less attracted than before, will ascend to a still greater distance, and so the ec centricity of the orbit will be increased still more. Therefore if the ratio of the increase and decrease of the centripetal force be augmented each revolution, the eccentricity will be augmented also ; and, on the contrary, if that ratio decrease, it will be diminished. Now, therefore, in the system of the bodies T, P, S, when the apsides of the orbit FAB are in the quadratures, the ratio of that increase and de crease is least of all, and becomes greatest when the apsides are in the syzygies. If the apsides are placed in the quadratures, the ratio near the apsides is less, and near the syzygies greater, than the duplicate ratio of the distances : and from that Greater ratio arises a direct motion of the line of 7 o the apsides, as was just now said. But if we consider the ratio of the whole increase or decrease in the progress between the apsides, this is less than the duplicate ratio of the distances. The force in the lower is to the force in the upper apsis in less than a duplicate ratio of the distance of the upper apsis from the focus of the ellipsis to the distance of the lower apsis from the same focus ; and, contrariwise, when the apsides are placed in the syzygies, the force in the lower apsis is to the force in the upper apsis in a greater than a duplicate ratio of the distances. For the forces LM in the quadratures added to the forces of the body T compose forces in a less ra tio ; and the forces KL in the syzygies subducted from the forces of the body T, leave the forces in a greater ratio. Therefore the ratio of the whole increase and decrease in the passage between the apsides is least at the quadratures and greatest at the syzygies ; and therefore in the passage of the apsides from the quadratures to the syzygies it is continually aug mented, and increases the eccentricity of the ellipsis ; and in the passage from the syzygies to the quadratures it is perpetually decreasing, and di minishes the eccentricity. COR. 10. That we may give an account of the errors as to latitude, let us suppose the plane of the orbit EST to remain immovable; and from the cause of the errors above explained, it is manifest, that, of the two forces NM, ML, which are the only and entire cause of them, the force ML acting always in the plane of the orbit PAB never disturbs the mo tions as to latitude ; and that the force NM, when the nodes are in the gyzygies, acting also in the same plane of the orbit, does not at that time affect those motions. But when the nodes are in the quadratures, it dis turbs tliem very much, and, attracting the body P perpetually out of the plane of its orbit, it diminishes the inclination of the plane in the passage of the body from the quadratures to the syzygies, and again increases the same in the passage from the syzygies to the quadratures. Hence it comes to pass that when the body is in the syzygies, the inclination is then least of all, and returns to the first magnitude nearly, when the body SEC. XL] OF NATURAL PHILOSOPHY. 209 arrives at the next node. But if the nodes are situate at the octants after the quadratures, that is, between C and A, D and B, it will appear, from ii C L E wnat was just now shewn, that in the passage of the body P from either node to the ninetieth degree from thence, the inclination of the plane is perpetually diminished ; then, in the passage through the next 45 degrees to the next quadrature, the inclination is increased ; and afterwards, again, in its passage through another 45 degrees to the next node, it is dimin ished. Therefore the inclination is more diminished than increased, and is therefore always less in the subsequent node than in the preceding one. And, by a like reasoning, the inclination is more increased than diminish ed when the nodes are in the other octants between A and D, B and C. The inclination, therefore, is the greatest of all when the nodes are in the syzygies In their passage from the syzygies to the quadratures the incli nation is diminished at each appulse of the body to the nodes : and be comes least of all when the nodes are in the quadratures, and the body in the syzygies ; then it increases by the same degrees by which it decreased before ; and, when the nodes come to the next syzygies, returns to its former magnitude. COR. 11. Because when the nodes are in the quadratures the body P is perpetually attracted from the plane of its orbit ; and because this attrac tion is made towards S in its passage from the node C through the con junction A to the node D ; and to the contrary part in its passage from the node D through the opposition B to the node C; it is manifest that, in its motion from the node C, the body recedes continually from the former plane CD of its orbit till it comes to the next node; and therefore at that node, being now at its greatest distance from the first plane CD, it will pass through the plane of the orbit EST not in D, the other node of that plane, but in a point that lies nearer to the body S, which therefore be comes a new place of the node in, antecedent ia to its former place. And, by a like reasoning, the nodes will continue to recede in their passage from this node to the next. The nodes, therefore, when situate in the quadratures, recede perpetually ; and at the syzygies, where no perturba tion can be produced in the motion as to latitude, are quiescent : in the in termediate places they partake of both conditions, and recede more slowly ; and, therefore, being always either retrograde or stationary, they will be carried backwards, or in atitecedentia, each revolution. COR. 12. All the errors described in these corrollaries arc a little greater 14 210 THE MATHEMATICAL PRINCIPLES BOOK L at the conjunction of the bodies P, S, than at their opposition ; because the generating forces NM and ML are greater. COR. 13. And since the causes and proportions of the errors and varia tions mentioned in these Corollaries do not depend upon the magnitude of the body S, it follows that all things before demonstrated will happen, if the magnitude of the body S be imagined so great as that the system of the two bodies P and T may revolve about it. And from this increase of the body S, and the consequent increase of its centripetal force, from which the errors of the body P arise, it will follow that all these errors, at equal dis tances, will be greater in this case, than in the other where the body S re volves about the system of the bodies P and T. COR. 14. But since the forces NM, ML, when the body S is exceedingly distant, are very nearly as the force SK and the ratio PT to ST con- junctly ; that is, if both the distance PT, and the absolute force of the body 8 be given, as ST3 reciprocally : and since those forces NM, ML are the causes of all the errors and effects treated of in the foregoing Corollaries; it is manifest that all those effects, if the system of bodies T and P con tinue as before, and only the distance ST and the absolute force of the body S be changed, will be very nearly in a ratio compounded of the direct ratio of the absolute force of the body S, and the triplicate inverse ratio of the distance ST. Hence if the system of bodies T and P revolve about a dis tant body S, those forces NM, ML, and their eifl ts, will be (by Cor. 2 and 6, Prop IV) reciprocally in a duplicate ratio c/f the periodical time. And thence, also, if the magnitude of the bodv S be proportional to its absolute force, those forces NM, ML, and their effects, will be directly as the cube of the apparent diameter of the distant body S viewed from T, and so vice versa. For these ratios are the same as the compounded ratio above men tioned. COR. 15. And because if the orbits ESE and PAB, retaining their fig ure, proportions, and inclination to each other, should alter their magni tude ; arid the forces of the bodies S and T should either remain, or be changed in any given ratio ; these forces (that is, the force of the body T, which obliges the body P to deflect from a rectilinear course into the orbit PAB, and the force of the body S, which causes the body P to deviate from that orbit) would act always in the same manner, and in the same propor tion : it follows, that all the effects will be similar and proportional, arid the times of those effects proportional also ; that is, that all the linear er rors will be as tne diameters of the orbits, the angular errors the same as before ; and the times of similar linear errors, or equal angular errors? as the periodical times of the orbits. COR. 16. Therefore if the figures of the orbits and their inclination to each other be given, and the magnitudes, forces, arid distances of the bodies he any how changed, we may. from the errors and times of those errors in SEC. XI.] OF NATURAL PHILOSOPHY. 2 \\ one case, collect very nearly the errors and times of the errors in any other case. But this may be done more expeditiously by the following method. The forces NM; ML, other things remaining unaltered, are as the radius TP ; and their periodical effects (by Cor. 2, Lein. X) are as the forces and the square of the periodical time of the body P conjunctly. These are the linear errors of the body P ; and hence the angular errors as they appear from the centre T (that is, the motion of the apsides and of the nodes, and all the apparent errors as to longitude and latitude) are in each revolution of the body P as the square of the time of the revolution, very nearly. Let these ratios be compounded with the ratios in Cor. 14, and in any system of bodies T, P, S, where P revolves about T very near to it, and T re volves about S at a great distance, the angular errors of the body P, ob served from the centre T, will be in each revolution of the body P as the square of the periodical time of the body P directly, and the square of the periodical time of the body T inversely. And therefore the mean motion of the line of the apsides will be in a given ratio to the mean motion of the nodes ; and both those motions will be as the periodical time of the body P directly, and the square of the periodical time of the body T in versely. The increase or diminution of the eccentricity and inclination of the orbit PAB makes no sensible variation in the motions of the apsides* and nodes, unless that inc/case or diminution be very great indeed. COR. 17. Sines the line LM becomes sometimes greater and sometimes less than the radius PT, let the mean quantity of the force LM be expressed E C sa— - -•::-.•.•::::::; by that radius PT ; and then that mean force will be to the mean force SK or SN (which may be also expressed by ST) as the length PT to the length ST. But the mean force SN or ST, by which the body T is re tained in the orbit it describes about S, is to the force with which the body P is retained in its orbit about T in a ratio compounded of the ratio of the radius ST to the radius PT, and the duplicate ratio of the periodical time of the body P about T to the periodical time of the body T about S. And, ex cequo, the mean force LM is to the force by which the body P is retain ed in its orbit about T (or by which the same body P might revolve at the distance PT in the same periodical time about any immovable point T) in the same duplicate ratio of the periodical times. The periodical times therefore being given, together with the distance PT, the mean force LM is also given ; and that force being given, there is given also the force MN, very nearly, by the analogy of the lines PT and MN. 212 THE MATHEMATICAL PRINCIPLES [BoOK I Con. IS. By tlie same laws by which the body P revolves about the body T, let us suppose many fluid bodies to move round T at equal dis tances from it ; and to be so numerous, that they may all become contiguous to each other, so as to form a fluid annul us, or ring, of a round figure, and concentrical to the body T; and the several parts of this annulus, perform ing their motions by the same law as the body P, will draw nearer to the body T, and move swifter in the conjunction and opposition of themselves and the body S, than in the quadratures. And the nodes of this annulus, or its intersections with the plane of the orbit of the body S or T, will rest at the syzygies ; but out of the syzygies they will be carried backward, or in. antecedentia ; with the greatest swiftness in the quadratures, and more slowly in other places. The inclination of this annulus also will vary, and its axis will oscillate each revolution, and when the revolution is completed will return to its former situation, except only that it will be carried round a little by the precession of the nodes. COR. 19. Suppose now the spherical body T, consisting of some matter not fluid, to be enlarged, and to extend its 'If on every side as far as that annulus, and that a channel were cut all round its circumference contain ing water j and that this sphere revolves uniformly about its own axis in the same periodical time. This water being accelerated and retarded by turns (as in the last Corollary), will be swifter at the syzygies, and slower at the quadratures, than the surface of the globe, and so will ebb and flow in its channel after the manner of the sea. If the attraction of the body S were taken away, the water would acquire no motion of flux and reflux by revolv- .ng round the quiescent centre of the globe. The case is the same of a globe moving uniformly forwards in a right line, and in the mean time revolving about its centre (by Cor. 5 of the Laws of Motion), and of a globe uni formly attracted from its rectilinear course (by Cor. 6, of the same Laws). But let the body S come to act upon it, and by its unequable attraction the A\ater will receive this new motion ; for there will be a stronger attraction upon that part of the water that is nearest to the body, and a weaker upon that part which is more remote. And the force LM will attract the water downwards at the quadratures, and depress it as far as the syzygies ; and the force KL will attract it upwards in the syzygies, and withhold its descent, and make it rise as far as the quadratures ; except only in so far as the motion of flux and reflux may be directed by the channel of the water, and be a little retarded by friction. COR. 20. If, now, the annulus becomes hard, and the globe is diminished, the motion of flux and reflux will cease ; but the oscillating motion of the inclination and the praecession of the nodes will remain. Let the globe have the same axis with the annulus, and perform its revolutions in the same times, and at its surface touch the annulus within, and adhere to it; then the globe partaking of the motion of the annulus, this whole compares SEC. XI. OF NATURAL PHILOSOPHY. 213 will oscillate, and the nodes will go backward, for the globe, as \ve shall shew presently, is perfectly indifferent to the receiving of all impressions. The greatest angle of the inclination of the annulus single is when the nodes are in the syzygies. Thence in the progress of the nodes to the quadratures, it endeavours to diminish its inclination, and by that endea vour impresses a motion upon the whole globe. The globe retains this motion impressed, till the annulus by a contrary endeavour destroys that motion, and impresses a new motion in a contrary direction. And by this means the greatest motion of the decreasing inclination happens when the nodes are in the quadratures; and the least angle of inclination in the octants B after the quadratures ; and, again, the greatest motion of roclination happens when the nodes are in the syzygies ; and the greatest angle of reclination in the octants following. And the case is the same of a globe without this an nulus, if it be a little higher or a little denser in the equatorial than in the polar regions : for the excess of that matter in the regions near the equator supplies the place of the annulus. And though we should suppose the cen tripetal force of this globe to be any how increased, so that all its parts were to tend downwards, as the parts of our earth gravitate to the centre, yet the phenomena of this and the preceding Corollary would scarce be al tered ; except that the places of the greatest and least height of the water will be different : for the water is now no longer sustained and kept in its orbit by its centrifugal force, but by the channel in which it flows. And, besides, the force LM attracts the water downwards most in the quadra tures, and the force KL or NM — LM attracts it upwards most in the syzygies. And these forces conjoined cease to attract the water downwards, and begin to attract it upwards in the octants before the syzygies ; and cease to attract the water upwards, and begin to attract the water down wards in the octants after the syzygies. And thence the greatest height of the water may happen about the octants after the syzygies ; and the least height about the octants after the quadratures ; excepting only so far as the motion of ascent or descent impressed by these forces may by the vis insita of the water continue a little longer, or be stopped a little sooner by impe diments in its channel. COR. 21. For the same reason that redundant matter in the equatorial regions of a globe causes the nodes to go backwards, and therefore by the increase of that matter that retrogradation is increased, by the diminution is diminished, and by the removal quite ceases : it follows, that, if more than 214 THE MATHEMATICAL PRINCIPLES [BOOK I that redundant matter be taken away, that is, if the globe be either more depressed, or of a more rare consistence near the equator than near the poles, there will arise a motion of the nodes in consequentia. COR. 22. And thence from the motion of the nodes is known the consti tution of the globe. That is, if the globe retains unalterably the same poles, and the motion (of the nodes) be in. antecedetitia, there is a redundance oi the matter near the equator; but if in conseqnentia, a deficiency. Sup pose a uniform and exactly spherical globe to be first at rest in a free space : then by some impulse made obliquely upon its superficies to be driven from its place, and to receive a motion partly circular and partly right forward. Because this globe is perfectly indifferent to all the axes that pass through its centre, nor has a greater propensity to one axis or to one situation oi the axis than to any other, it is manifest that by its own force it will never change its axis, or the inclination of it. Let now this globe be impelled obliquely by a new impulse in the same part of its superficies as before . and since the effect of an impulse is not at all changed by its coming sooner or later, it is manifest that these two impulses, successively impressed, will produce the same motion as if they were impressed at the same time : that is, the same motion as if the globe had been impelled by a simple force compounded of them both (by Cor. 2, of the Laws), that is, a simple motion about an axis of a given inclination. And the case is the same if the sec ond impulse were made upon any other place of the equator of the first motion ; and also if the first impulse were made upon any place in the equator of the motion which would be generated by the second impulse alone; and therefore, also, when both impulses are made in any places whatsoever ; for these impulses will generate the same circular motion as if they were impressed together, and at once, in the place of the intersec tions of the equators of those motions, which would be generated by each of them separately. Therefore, a homogeneous and perfect globe will not retain several distinct motions, but will unite all those that are impressed on it, and reduce them into one; revolving, as far as in it lies, always with a simple and uniform motion about one single given axis, with an inclina tion perpetually invariable. And the inclination of the axis, or the velocity of the rotation, will not be changed by centripetal force. For if the globe be supposed to be divided into two hemispheres, by any plane whatsoever passing through its own centre, and the centre to which the force is direct ed, that force will always urge each hemisphere equally ; and therefore will not incline the globe any way as to its motion round its own axis. But let there be added any where between the pole and the equator a heap oi new matter like a mountain, and this, by its perpetual endeavour to recede from the centre of its motion, will disturb the motion of the globe, and cause its poles to wander about its superficies, describing circles about themselves and their opposite points. Neither can this enormous evagatior XL] OF NATURAL PHILOSOPHY. 2 In of the poles be corrected, unless by placing that mountain ei '. er in one ol the poles; in which case, by Cor. 21, the nodes of the equator will go for wards ; or in the equatorial regions, in which case, by Cor. 20, the nodes will go backwards: or, lastly, by adding on the other side of the axis anew quantity of matter, by which the mountain may be balanced in its motion; and then the nodes will either go forwards or backwards, as the mountain and this newly added matter happen to be nearer to the pole or to the equator. PROPOSITION LXV1I. THEOREM XXVII. The same laics of attraction being' supposed, I say, that the exterior body S does, by radii dra.cn to the point O, the common centre of gravity of the interior bodies P and T, describe round that centre areas more proportional to the times, and an orbit more approaching to the form of an ellipsis having its focus in that cen > •.-. than, it can describe round the innermost and greatest body T by ra Hi drawn to that body. For the attractions of the body S towards T and P compose its absolute attraction, which is more directed towards O, the common centre of gravity S(i of the bodies T and P, than it is to the «. reatest body T ; and which is more in a reciprocal propor tion to the square of the distance SO, than it is to the square of the distance ST : as will easily appear by a little consideration. PROPOSITION LXVIII. THEOREM XXVIII. The same laws of attraction supposed, I say, that the exterior body S will, by radii drawn to O, the common centre of gravity of the interior bodies P and T, describe round that centre areas more propor tional to the times, and an orbit more approaching to the form of an ellipsis having its focus in that centre, if the innermost and greatest body be agitated by these attractions as well as the rest, than it would do if that body were either at rest as not attracted, or were much tnore or much less attracted, or much more or much less agitated. This may be demonstrated after the same manner as Prop. LXVI, but by a more prolix reasoning, which I therefore pass over. It will be suf ficient to consider it after this manner. From the demonstration of the last Proposition it is plain, that the centre, towards which the body S is urged by the two forces conjunctly, is very near to the common centre of gravity of those two other bodies. If this centre were to coincide with that common centre, and moreover the common centre of gravity of all the three bodies were at rest, the body S on one side, and the common centre of gravity of the other two bodies on the other side, would describe true ellip* 216 THE MATHEMATICAL PRINCIPLES - [BOOK 1 ses about that quiescent common centre. This appears from Cor. 2, Pro]) LVIII, compared with what was demonstrated in Prop. LX1V, and LXY Now this accurate elliptical motion will be disturbed a little by the dis tance of the centre of the two bodies from the centre towards which tht third body S is attracted. Let there be added, moreover, a motion to the Bommon centre of the three, and the perturbation will be increased yet more. Therefore the perturbation is least when the common centre of the three bodies is at rest; that I is, when the innermost and greatest body T is at tracted according to the same law as the rest are ; and is always greatest when the common centre of the three, by the diminution of the motion of the body T, begins to be moved, and is more and more agitated. COR. And hence if more lesser bodies revolve about the great one, it may easily be inferred that the orbits described will approach nearer to ellipses ; and the descriptions of areas will be more nearly equable, if all the bodies mutually attract and agitate each other with accelerative forces that are as their absolute forces directly, and the squares of the distances inversely : and if the focus of each orbit be placed in the common centre of gravity of all the interior bodies (that is. if the focus of the first and in nermost orbit be placed in the centre of gravity of the greatest and inner most body : the focus of the second orbit in the common centre of gravity of the two innermost bodies; the focus of the third orbit in the common centre of gravity of the three innermost ; and so on), than if the innermost body were at rest, and was made the common focus of all the orbits. PROPOSITION LXIX. THEOREM XXIX. fn a system of several bodies A, B, C, D, $*c., if any one of those bodies, as A, attract all the rest, B, C, D, $*c.,with accelerative forces that are reciprocally as the squares of the distances from the attracting body ; and another body, as B, attracts also the rest. A, C, D, $-c., with forces that are reciprocally as the squares of the distances from the attract ing body ; the absolute forces of the attracting bodies A and B will be to each other as those very bodies A and B to which those forces belong. For the accelerative attractions of all the bodies B, C, D, towards A, are by the supposition equal to each other at equal distances ; and in like manner the accelerative attractions of all the bodies towards B are also equal to each other at equal distances. But the absolute attractive force of the body A is to the absolute attractive force of the body B as the ac- eelerative attraction of all the bodies towards A to the accelerative attrac tion of all the bodies towards B at equal distances ; and so is also the ac celerative attraction of the body B to*vards A to the accelerative attraction SEC. XI] OF NATURAL PHILOSOPHY. 21 T of the body A towards B. But the accelerative attraction of the body B towards A is to the accelerative attraction of the body A towards B as the mass of the body A to the mass of the body B ; because the motive forces which (by the 2d, 7th, and 8th Definition) are as the accelerative forces and the bodies attracted conjunctly are here equal to one another by the third Law. Therefore the absolute attractive force of the body A is to the absolute attractive force of the body B aa the mass of the body A to the mass of the body B. Q.E.D. COR. 1. Therefore if each of the bodies of the system A, B, C, D, &c. does singly attract all the rest with accelerative forces that are reciprocally as the squares of the distances from the attracting body, the absolute forces of all those bodies will be to each other as the bodies themselves. COR. 2. By a like reasoning, if each of the bodies of the system A, B, C, D, &c., do singly attract all the rest with accelerative forces, which are either reciprocally or directly in the ratio of any power whatever of the distances from the attracting body : or which are defined by the distances from each of the attracting bodies according to any common law : it is plain that the absolute forces of those bodies are as the bodies themselves. COR. 3. In a system of bodies whose forces decrease in the duplicate ra tio of the distances, if the lesser revolve about one very great one in ellip ses, having their common focus in the centre of that great body, and of a figure exceedingly accurate ; and moreover by radii drawn to that great ody describe areas proportional to the times exactly • the absolute forces )i those bodies to each other will be either accurately or very nearly in the ratio of the bodies. And s > on the contrary. This appears from Cor. of Prop. XLVII1, compared with the first Corollary of this Prop. SCHOLIUM. These Propositions naturally lead us to the analogy there is between centripetal forces, and the central bodies to which those forces used to be directed ; for it is reasonable to suppose that forces which are directed to bodies should depend upon the nature and quantity of those bodies, as we see they do in magnetical experiments. And when such cases occur, we are to compute the attractions of the bodies by assigning to each of their particles its proper force, and then collecting the sum of them all. I here ue*e the word attraction in general for any endeavour, of what kind soever, made by bodies to approach to each other; whether that endeavour arise from the action of the bodies themselves, as tending mutually to or agita ting each other by spirits emitted; or whether it arises from the action of the aether or of the air, or of any medium whatsoever* whether corporeal or incorporeal, any how impelling bodies placed therein towards each other. In the same general sense I use the word impulse, not defining in this trea tise the species or physical qualities of forces, but investigating the quantities THE MATHEMATICAL PRINCIPLES [BOOK ). and mathematical proportions of them ; as I observed before ir (lie Defi nitions. In mathematics we are to investigate the quantities of forces with their proportions consequent upon any conditions supposed ; then, when we enter upon physics, we compare those proportions with the phe nomena of Nature, that we may know what conditions of those forces an swer to the several kinds of attractive bodies. And this preparation being made, we argue more safely concerning the physical species, causes, and proportions of the forces. Let us see, then, with what forces spherical bodies consisting of particles endued with attractive powers in the manner above spoken of must act mutually upon one another : and what kind of motions will follow from thence. SECTION XII. Of the attractive forces of sphcerical bodies. PROPOSITION LXX. THEOREM XXX. If to every point of a spherical surface there tend equal centripetal forces decreasing in, the duplicate ratio of the distances from those points ; I say, that a corpuscle placed within that superficies will not be attract ed by those forces any way. Let HIKL, be that sphaerical superficies, and P a corpuscle placed within. Through P let there be drawn to this superficies to two lines HK, IL, inter- cepting very small arcs HI, KL ; and because (by Cor. 3, Lem. VII) the triangles HPI,LPK are alike, those arcs will be proportional to the distances HP LP ; and any particles at HI and KL of the spheri cal superficies, terminated by right lines passing through P, will be in the duplicate ratio of those distances. Therefore the forces of these particles exerted upon the body P are equal between themselves. For the forces are as the particles directly, and the squares of the distances inversely. And these two ratios compose the ratio of equality. The attractions therefore, being made equally towards contrary parts, destroy each other. And by a like reasoning all the attractions through the whole spherical superficies are destroyed by contrary attractions. Therefore the body P will not be any way impelled by those attractions. Q.E.D. PROPOSITION LXXI. THEOREM XXXI. The same things supposed as above, I say, that a cor pu vie placed with out the sph(eric»l superficies is attracted towards the centre of tht sphere wiih a force reciprocally proportional to the square of its dis tance from that centre. Let AHKB, ahkb, be two equal sphaerical superficies described about SEC. XII.J OF NATURAL PHILOSOPHY. the centre S, s ; their diameters AB, ab ; and let P and p be two corpus cles situate without the gpheres in those diameters produced. Let there be drawn from the corpuscles the lines PHK, PIL, phk, pil, cutting off from the great circles AHB, ahb, the equal arcs HK, hk, IL; il ; and to those lines let fall the perpendiculars SD, sd, SE, SP, 1R, ir ; of which let SD, sd, cut PL, pi, in F and f. Let fall also to the diameters the perpen diculars IQ, iq. Let now the angles DPE, dpe, vanish; and because DS and ds, ES and es are equal, the lines PE, PP, and pe, pf, and the lineolso I )F, df may be taken for equal ; because their last ratio, when the angles DPE, dpe vanish together, is the ratio of equality. These things then supposed, it will be, as PI to PF so is RI to DF, and as pf to pi so is df or DF to ri ; and, ex cequo, as PI X pf to PF X pi so is R I to ri, that is (by Cor. 3, Lem VII), so is the arc IH to the arc ih. Again, PI is to PS as IQ. to SE, and ps to pi as se or SE to iq ; and, ex ceqno, PI X ps to PS X pi as IQ. to iq. And compounding the ratios PI2 X pf X ps is to pi2 X PF X PS, as IH X IQ to ih X iq ; that is, as the circular super ficies which is described by the arc IH, as the semi-circle AKB revolves about the diameter AB, is to the circular superficies described by the arc ih as the semi-circle akb revolves about the diameter ab. And the forces with which these superficies attract the corpuscles P and p in the direction of lines tending to those superficies are by the hypothesis as the superficies themselves directly, and the squares of the distances of the superficies from those corpuscles inversely; that is, as pf X ps to PF XPS. And these forces again are to the oblique parts of them which (by the resolution of forces as in Cor. 2, of the Laws) tend to the centres in the directions of the lines PS, JDS-, as PI to PQ, and pi to pq ; that is (because of the like trian gles PIQ and PSF, piq and psf\ as PS to PF and ps to pf. Thence ex cequO) the attraction of the corpuscle P towards S is to the attraction of PF XpfXps. pf X PF X PS . the corpusclejo towards 5 as ~ =— is to — , that is, as ps2 to PS2. And, by a like reasoning, the forces with which the su perficies described by the revolution of the arcs KL, kl attract those cor puscles, will be as jDS2 to PS2. And in the same ratio will be the foroes of all the circular superficies into which each of the sphaerical superficies may be divided by taking sd always equal to SD, and se equal to SE. And therefore, by composition, the forces of the entire spherical superficies ex erted upon those corpuscles will be in the same ratio. Q.E.D 220 THE MATHEMATICAL PRINCIPLES [BOOK 1 PROPOSITION LXXIL THEOREM XXXII. If to the several points of a sphere there tend equal centripetal forces de creasing in a duplicate ratio of the distances from those points ; and there be given both the density of the sphere and the ratio of the di ameter of the sphere to the distance of the corpuscle from its centre ; I say, that the force with which the corpuscle is attracted is propor tional to the semi-diameter of the sphere. For conceive two corpuscles to be severally attracted by two spheres, one by one, the other by the other, and their distances from the centres of the spheres to be proportional to the diameters of the spheres respectively , and the spheres to be resolved into like particles, disposed in a like situation to the corpuscles. Then the attractions of one corpuscle towards the sev eral particles of one sphere will be to the attractions of the other towards as many analogous particles of the other sphere in a ratio compounded of the ratio of the particles directly, and the duplicate ratio of the distances inversely. But the particles are as the spheres, that is, in a triplicate ra tio of the diameters, and the distances are as the diameters ; and the first ratio directly with the last ratio taken twice inversely, becomes the ratio of diameter to diameter. Q.E.D. COR. 1. Hence if corpuscles revolve in circles about spheres composed of matter equally attracting, and the distances from the centres of the spheres be proportional to their diameters, the periodic times will be equal. COR. 2. And, vice versa, if the periodic times are equal, the distances will be proportional to the diameters. These two Corollaries appear from Cor. 3, Prop. IV. COR. 3. If to the several points of an^ two solids whatever, of like fig- are and equal density, there tend equal centripetal forces decreasing in a duplicate ratio of the distances from those points, the forces, with which corpuscles placed in a like situation to those two solids will be attracted by them, will be to each other as the diameters of the solids. PROPOSITION LXXIII. THEOREM XXXIII. If to the several points of a given sphere there tend equal centripetal forces decreasing in a duplicate ratio of the distances from the points ; 1 say, that a corpuscle placed within the sphere is attracted by a force proportional to its distance from the centre. In the sphere ABCD, described about the centre S, let there be placed the corpuscle P ; and about the same centre S, with the interval SP? conceive de- |B scribed an interior sphere PEQP. It is plain (by Prop. LXX) that the concentric sphaerical superficies, of which the difference AEBF of the spheres is com posed, have no effect at all upon the body P, their at- SEC. XIL] OF NATURAL PHILOSOPHY. 22\ tractions being destroyed by contrary attractions. There remains, there fore; only the attraction of the interior sphere PEQ,F. And (by Prop. LXXII) this is as the distance PS. Q.E.D. SCHOLIUM. By the superficies of which I here imagine the solids composed, I do not mean superficies purely mathematical, but orbs so extremely thin, that their thickness is as nothing; that is, the evanescent orbs of which the sphere will at last consist when the number of the orbs is increased, and their thickness diminished without end. In like manner, by the points of which lines, surfaces, and solids are said to be composed, are to be understood equal particles, whose magnitude is perfectly inconsiderable. PROPOSITION LXXIV. THEOREM XXXIV. The same things supposed, I say, that a corpuscle situate without the sphere is attracted with a force reciprocally proportional to the square of its distance from the centre. For suppose the sphere to be divided into innumerable concentric sphe rical superficies, and the attractions of the corpuscle arising from the sev eral superficies will be reciprocally proportional to the square of the dis tance of the corpuscle from the centre of the sphere (by Prop. LXXI). And, by composition, the sum of those attractions, that is, the attraction of the corpuscle towards the entire sphere, will be in the same ratio. Q.E.D. COR. 1. Hence the attractions of homogeneous spheres at equal distances from the centres will be as the spheres themselves. For (by Prop. LXXII) if the distances be proportional to the diameters of the spheres, the forces will be as the diameters. Let the greater distance be diminished in that ratio ; and the distances now being equal, the attraction will be increased in the duplicate of that ratio ; and therefore will be to the other attraction in the triplicate of that ratio ; that is, in the ratio of the spheres. COR. 2. At any distances whatever the attractions are as the spheres applied to the squares of the distances. COR. 3. If a corpuscle placed without an homogeneous sphere is attract ed by a force reciprocally proportional to the square of its distance from the centre, and the sphere consists of attractive particles, the force of ever y particle will decrease in a duplicate ratio of the distance from each particle. PROPOSITION LXXV. THEOREM XXXV. If to the several points of a given sphere there tend equal centripetal forces decreasing in a duplicate ratio of the distances from the points ; I say, that another similar sphere will be attracted by it with a force recip rocally proportional to the square of the distance of the centres. For the attraction of every particle is reciprocally as the square of its 222 THE MATHEMATICAL PRINCIPLES | BOOK L distance from the centre of the attracting sphere (by Prop. LXXIV). and is therefore the same as if that whole attracting force issued from one sin gle corpuscle placed in the centre of this sphere. But this attraction is as great as on the other hand the attraction of the same corpuscle would be, if that were itself attracted by the several particles of the attracted sphere with the same force with which they are attracted by it. But that attrac tion of the corpuscle would be (by Prop. LXXIV) reciprocally propor tional to the square of its distance from the centre of the sphere : therefore the attraction of the sphere, equal thereto, is also in the same ratio. Q,.E.D. COR. 1. The attractions of spheres towards other homogeneous spheres are as the attracting spheres applied to the squares of the distances of their centres from the centres of those which they attract. COR. 2. The case is the same when the attracted sphere does also at tract. For the several points of the one attract the several points of the other with the same force with which they themselves are attracted by the others again; and therefore since in all attractions (by Law III) the at tracted and attracting point are both equally acted on, the force will be doubled by their mutual attractions, the proportions remaining. COR. 3. Those several truths demonstrated above concerning the motion of bodies about the focus of the conic sections will take place when an attracting sphere is placed in the focus, and the bodies move without the sphere. COR. 4. Those things which were demonstrated before of the motion of bodies about the centre of the conic sections take place when the motions are performed within the sphere. PROPOSITION LXXVI. THEOREM XXXVI. ff spheres be however dissimilar (as to density of matter and attractive, force] in the same ratio onward from the centre to the circumference ; but every where similar, at every given distance from the centre, on all sides round about ; and the attractive force of every point decreases in the duplicate ratio of the distance of the body attracted ; I say, that the whole force with which one of these spheres attracts the oilier will be reciprocally proportional to the square of the distance of the centres. Imagine several concentric similar spheres, AB, CD, EF, &c.. the inner most of which added to the outermost may compose a matter more dense to wards the centre, or subducted from them may leave the same more lax and rare. Then, by Prop. LXXV, these sphere? will attract other similar con- SEC. XII.] OF NATURAL PHILOSOPHY. 223 eentric spheres GH; IK, LM, &c., each the other, with forces reciprocally proportional to the square of the distance SP. And, by composition or division, the sum of all those forces, or the excess of any of them above the others; that is, the entire force with which the whole sphere AB (com posed of any concentric spheres or of their differences) will attract the whole sphere GH (composed of any concentric spheres or their differences) in the same ratio. Let the number of the concentric spheres be increased in infinitum, so that the density of the matter together with the attractive force may, in the progress from the circumference to the centre, increase or decrease according to any given law ; and by the addition of matter not at tractive, let the deficient density be supplied, that so the spheres may acquire any form desired ; and the force with which one of these attracts the other will be still, by the former reasoning, in the same ratio of the square of the distance inversely. Q.E.I). COR. I. Hence if many spheres of this kind, similar in all respects, at tract each other mutually, the accelerative attractions of each to each, at any equal distances of the centre's, will be as the attracting spheres. COR. 2. And at any unequal distances, as the attracting spheres applied to the squares of the distances between the centres. /' COR. 3. The motive attractions, or the weights of the spheres towards one another, will be at equal distances of the centres as the attracting and attracted spheres conjunctly ; that is, as the products arising from multi plying the spheres into each other. COR. 4. And at unequal distances, as those products directly, and the squares of the distances between the centres inversely. COR. 5. These proportions take place also when the attraction arises from the attractive virtue of both spheres mutually exerted upon each other. For the attraction is only doubled by the conjunction of the forces, the proportions remaining as before. COR. 6. If spheres of this kind revolve about others at rest, each about each ; and the distances between the centres of the quiescent and revolving bodies are proportional to the diameters of the quiescent bodies ; the peri odic times will be equal. COR. 7. And, again, if the periodic times are equal, the distances will be proportional to the diameters. COR. 8. All those truths above demonstrated, relating to the motions 'jf bodies about the foci of conic sections, will take place when an attract ing sphere, of any form and condition like that above described, is placed in the focus. COR. 9. And also when the revolving bodies are also attracting spheres Df any condition like that above described. 224 THE MATHEMATICAL PRINCIPLES [BOOK I. PROPOSITION LXXVI1. THEOREM XXXVII. Tf to 1 he several points of spheres there tend centripetal forces propor tional to the distances of the points from the attracted bodies ; I say, that the compounded force with which two spheres attract each other mutually is as the distance between the centres of the spheres. CASE 1. Let AEBF be a sphere ; S its centre . P a corpuscle attracted : PA SB the axis of the sphere passing through the centre of the corpuscle ; EF, ef two planes cutting the sphere, and perpendicular to the axis, and equi-distant, one on one side, the other on the other, from the centre of the sphere ; G and g- the intersections of the planes and the axis ; and H any point in the plane EF. The centri petal force of the point H upon the corpuscle P, exerted in the direction of the line PH, is as the distance PH ; and (by Cor. 2, of the Laws) the same exerted in the direction of the line PG, or towards the . centre S, is as the length PG. Therefore the force of all the points in the plane EF (that is, of that whole plane) by which the corpuscle P is attracted towards the centre S is as the distance PG multiplied by the number of those points, that is, as the solid contained under that plane EF and the distance PG. And in like manner the force of the plane ef, by which the corpuscle P is attracted towards the centre S, is as that plane drawn into its distance Pg, or as the equal plane EF drawn into that distance Pg* ; and the sum of the forces of both planes as the plane EF drawn into the sum of the distances PG + P^, that is, as that plane drawn into twice the distance PS of the centre and the corpuscle ; that is, as twice the plane EF drawn into the dis tance PS, or as the sum of the equal planes EF + ef drawn into the same distance. And, by a like reasoning, the forces of all the planes in the whole sphere, equi-distant on each side from the centre of the sphere, are as the sum of those planes drawn into the distance PS, that is, as the whole sphere and the distance PS conjunctly. Q,.E.D. CASE 2. Let now the corpuscle P attract the sphere AEBF. And, by the same reasoning, it will appear that the force with which the sphere is attracted is as the distance PS. Q,.E.D. CASE 3. Imagine another sphere composed of innumerable corpuscles P : and because the force with which every corpuscle is attracted is as the dis tance of the corpuscle from the centre of the first sphere, and as the same sphere conjunctly, and is therefore the same as if it all proceeded from a single corpuscle situate in the centre of the sphere, the entire force with which all the corpuscles in the second sphere are attracted, that is, with which that whole sphere is attracted, will be the same as if that sphere SEC. Xll.] OP NATURAL PHILOSOPHY. 225 were attracted by a force issuing from a single corpuscle in the centre of the first sphere ; and is therefore proportional to the distance between the centres of the spheres. Q,.E.D. CASE 4. Let the spheres attract each other mutually, and the force will be doubled, but the proportion will remain. Q..E.D. CASE 5. Let the corpuscle p be placed within ^- ^\E the sphere AEBF ; and because the force of the plane ef upon the corpuscle is as the solid contain ed under that plane and the distance jog' ; and the contrary force of the plane EF as the solid con tained under that plane and the distance joG ; the ^ force compounded of both will be as the difference ** of the solids, that is, as the sum of the equal planes drawn into half the difference of the distances ; that is, as that sum drawn into joS, the distance of the corpuscle from the centre of the sphere. And, by a like reasoning, the attraction of all the planes EF, ef, throughout the whole sphere, that is, the attraction of the whole sphere, is conjunctly as the sum of all the planes, or as the whole sphere, and as joS, the distance of the corpuscle from the centre of the sphere. Q.E.D. CASE 6. And if there be composed a new sphere out of innumerable cor puscles such as jo, situate within the first sphere AEBF, it may be proved, as before, that the attraction, whether single of one sphere towards the other, or mutual of both towards each other, will be as the distance joS of the centres. Q, E.D. PROPOSITION LXXVIII. THEOREM XXXVIII. If spheres it* the progress from the centre to the circumference be hoivMtv dissimilar a->id unequable, but similar on every side round about af all given distances from the centre ; and the attractive force of evsrt/ point be as the distance of the attracted body ; I say, that the entire force with which two spheres of this kind attract each other mutitallij is proportional to the distance between the centres of the spheres. This is demonstrated from the foregoing Proposition, in the same man ner as Proposition LXXVI was demonstrated from Proposition LXXY. COR. Those things that were above demonstrated in Prop. X and LXJV, of the motion of bodies round the centres of conic sections, take place when all the attractions are made by the force of sphaerical bodies of the condi tion above described, and the attracted bodies are spheres of the same kind. SCHOLIUM. i have now explained the two principal cases of attractions; to wit, when the centripetal forces decrease in a duplicate ratio of the distances •r increase in a simple ratio of the distances, causing the bodies in botli 15 226 THE MATHEMATICAL PRINCIPLES [BoOK 1 cases to revolve in conic sections, and composing sphaerical bodies whose centripetal forces observe the same law of increase or decrease in the recess from the centre as the forces of the particles themselves do ; which is verv remarkable. It would be tedious to run over the other cases, whose con clusions are less elegant and important, so particularly as I have done these. I choose rather to comprehend and determine them all by one gen eral method as follows. LEMMA XXIX. ff about the centre S there be described any circle as AEB, and about the centre P there be. also described two circles EF, ef, cutting the Jirst in E and e, and the line PS in F and f ; and there be let fall to PS the perpendiculars ED, ed ; I say, that if the distance of the arcs EF; ef be supposed to be infinitely diminished, the last ratio of the evanscent linr Dd to the evanescent line Ff is the same as that of the line PE to the live PS. For if the line Pe cut the arc EF in q ; and the right line Ee, which coincides with the evanescent arc Ee, be produced, and meet the right line PS in T ; and there be let fall from S to PE the perpendicular SG ; then, because of the like triangles DTE, </'!>, DES, it will be as Dd to Ee so ))T to TE, or DE to ES : and because the triangles, Ee?, ESG (by Lem. VIII, and Cor. 3, Lem. VII) are similar, it will be as Ee to eq or F/soES to SG ; and, ex ceqno, as Dd to Ff so DE to SG ; that is (because of the similar triangles PDE; PGS), so is PE to PS. Q.E.D. PROPOSITION LXXIX. THEOREM XXXIX. Suppose a superficies as EFfe to have its breadth infinitely diminished, and to be just vanishing ; and that the same superficies by its revolu- tion round the axis PS describes a spherical concavo-convex solid, to the several equnJ particle* of which there tend equal centripetal forces ; I soy, that the force with which thit solid attracts a corpuscle situate in P is in a ratio compounded of the ratio of the solid DE2 X Ff and the ratio of the force with which the given particle in the place Ff would attract the same corpuscle. For if we consider, first, the force of the spherical superficies FE which SEC. xn.j OF NATURAL PHILOSOPHY. 227 is generated by the revolution of the arc FE, and is cut any where, as in r, by the line</6, the annular part of the super J'cies generated by the revolution of the arc rE will be as the lineola Dd, the radius of the sphere PE re- mainiag the same; as Archimedes has de monstrated in his Book of the Sphere and Cylinder. And the force of this super ficies exerted in the direction of the lines PE or Pr situate all round in the conical superficies, will be as this annular superficies itself; that is as the lineola DC/, or, which is the same, as the rectangle under the given radius PE of the sphere and the lineola DC/ ; but that force, exerted in the direction of the line PS tending to the centre S, will be less in the ratio PI) to PE, and therefore will be as PD X DC/. Suppose now the line DF to be divided into innumerable little equal par ticles, each of which call DC/, and then the superficies FE will be divided into so many equal annuli, whose forces will be as the sum of all the rec tangles PD X DC/, that is, as |PF2 — - |PD2; and therefore as DE-. Let now the superficies FE be drawn into the altitude F/; and the force of the solid EF/e exerted upon the corpuscle P will be as DE2 X Ff; that is, if the force be given which any given particle as Ff exerts upon the corpuscle P at the distance PF. But if that force be not given, the force of the solid EF/e will be as the solid DE2 X Ff and that force not given, conjunctly. Q.E.D. PROPOSITION LXXX. THEOREM XL. If to the several equal parts of a sphere ABE described about the centre S there tend equal centripetal forces ; and from the several points I) in the axis of the sphere AB in which a corpuscle, as F, is placed, there be erected the perpendiculars DE meeting the sphere in E, and if in those perpendiculars the lengths DN be taken as the quantity DE2 X PS — —-, , and as th* force which a particle of the sphere situate in, the axis exerts at the distance PE upon the corpuscle P conjunctly ; ] say, that the in hole force with which the, corpuscle P is attracted to wards the sphere is as the area ANB, comprehended under the axis of the sphere AB, and the curve line ANB, the locus of the point N. For supposing the construction in the last Lemma and Theorem to stand, conceive the axis of the sphere AB to be divided into innumerable equal particles DC/, and the whole sphere to be divided into so many sphe rical concavo-convex laminae EF/e / and erect the perpendicular dn. By the last Theorem, the force with which the laminas EF/e attracts the cor puscle P is as DE2 X Ff and the force of one particle exerted at the 228 THE MATHEMATICAL PRINCIPLES [BOOK I. distance PE or PF, conjunctly. But (by the last Lemma) Dd is to F/ as PE to PS, and therefore F/ . is equal to PE F/ is equal to Dd X ; and DE2 X DE2 X PS PET~; and therefore the force of the la- DE2 X PS mina EF/e is as Do? X PT?~ and the force of a particle exerted at the distance PF conjunctly ; that is, by the supposition, as DN X D(/7 or as the evanescent area DNwrf. Therefore the forces of all the lamina) exerted upon the corpuscle P are as all the areas DN//G?, that is, the whole force of the sphere will be as the whole area ANB. Q.E.D. COR. 1. Hence if the certripetal force tending to the several particles •p)F2 vx po remain always the same at all distances, and DN be made as ™ ; Jr Jli the whole force with which the corpuscle is attracted by the sphere is as the area ANB. COR. 2. If the centripetal force of the particles be reciprocally as the DE2 X PS distance of the corpuscle attracted by it, and DN be made as -— ^^ , the force with which the corpuscle P is attracted by the whole sphere wil] be as the area ANB. Cor. 3. Jf the centripetal force of the particles be reciprocally as the cube of the distance of the corpuscle attracted by it, and DN be made as T)F2 y PS — --- — — . the force with which the corpuscle is attracted by the whole sphere will be as the area ANB. COR. 4. And universally if the centripetal force tending to the several particles of the sphere be supposed to be reciprocally as the quantity V ; DE2 X PS and D5& be made as „„ — ^- ; the force with which a corpuscle is at- Jr Jtj X ' tracted by the whole sphere will be as the area ANB. PROPOSITION LXXXI. PROBLEM XLI. T/Le things remaining as above, it is required lo measure the area ANB. From the point P let there be drawn the right line PH touching the sphere in H ; and to the axis PAB, letting fall the perpendicular HI, bisect PI in L; and (by Prop. XII, Book II, El em.) PE2 is equal tf SEC. XII.] OF NATURAL PHILOSOPHY. 229 PS3 + SE2 + 2PSD. But because the triangles SPH, SHI are alike, SE2 or SH2 is equal to the rectan gle PSI, Therefore PE2 is equal to the rectangle contained under PS and PS -f SI + 2SD ; that is, under PS and 2LS + 2SD ; that is, under PS and 2LD. Moreover DE2 is equal to SE2 — SD% or SE2 — LS2 + 2SLD — LD2, that is, 2SLD — LD2 — ALB. For LS°- — SE2 or LSa — SAa (by Prop. VI, Book II, Elem.) is equal to the rectan gle ALB. Therefore if instead of DE 2 we write 2SLD — LD 2 — ALB, the quantity -™ - ^-, which (by Cor. 4 of the foregoing Prop.) is as PE x the length of the ordinate DN, will 2SLD x PS LD2 X PS now resolve itself into three parts ALB xPS ... -TE3rr~ -pfixT" -pE^-v-;whereifinsteadofVwewnt the inverse ratio of the centripetal force, and instead of PE the mean pro portional between PS and 2LD, those three parts will become ordinates to so many curve lines, whose areas are discovered by the common methods. Q.E.D. EXAMPLE 1. If the centripetal force tending to the several particles of the sphere be reciprocally as the distance ; instead of V write PE the dis tance, then 2PS X LD for PE2 ; and DN will become as SL — £ LD — •ny |y Suppose DN equal to its double 2SL — LD — - — r^ 5 an<* 2SL the given part of the ordinate drawn into the length AB will describe the rectangular area 2SL X AB ; and the indefinite part LD, drawn perpen dicularly into the same length with a continued motion, in such sort as in its motion one way or another it may either by increasing or decreasing re- LB2-LA2 main always equal to the length LD, will describe the area — — ^ — — , that is, the area SL X AB ; which taken from the former area 2SL X AB, leaves the area SL X AE. But the third part - ---, drawn after the i lit, same manner with a continued motion perpendicularly into the same length, will describe the area of an hyperbola, which subducted from the area SL X AB will leave ANB the area sought. Whence arises this construction of the Problem. At the points, L, A, B, erect the perpendiculars L/, Act, B6; making Aa equal to LB, and Bb equal to LA. Making L/ and LB asymptotes, describe through the points a, 6, 230 THE MATHEMATICAL PRINCIPLES [BOOK 1 the hyperbolic crrve ab. And the chord ba being drawn, will inclose the area aba equal to the area sought ANB. EXAMPLE 2. If the centripetal force tending to the several particles of the sphere be reciprocally as the cube of the distance, or (which is the same PE3 thing; as that cube applied to any given plane ; write 2PS X LD for PE2 ; and DN will become as '2AS2 SL X AS2 for V, and AS2 ALB X AS2 2PS X LD2 LSI PS X LD 2PS that is (because PS, AS, SI are continually proportional), as ALB X SI 2LD: LSI If we draw then these three parts into th length AB, the first r-pr will generate the area of an hyperbola ; the sec- L-t \J , ALB X SI . ALB X SI ond iSI the area } AB X SI ; the third 2Ll^ — area - 2LA , that is, !AB X SI. From the first subduct the sum of the 2LB second and third, and there will remain ANB, the area sought. Whence arises this construction of the problem. At the points L, A, S, B, erect the perpendiculars L/ Aa Ss, Bb, of which suppose Ss equal to SI ; and through the point s, to the asymptotes L/, LB, describe the hyperbola asb meeting the perpendiculars Aa, Bb, in a and b ; and the rectangle 2ASI, subducted from the hyberbolic area AasbB, will B leave ANB the area sought. .. ,, . „ EXAMPLE 3. If the centripetal force tending to the several particles of the spheres decrease in a quadruplicate ratio of the distance from the par- pT^4 _ tides ; write ~|f- for V, then V 2PS + LD for PE, and DN will become ___ V2SI X SI2 X ALB 2v2SI X These three parts drawn into the length AB, produce so many areas, viz. J-L 2SI2 X SL . 1 x^ — into T— r LA ~~~5ot in*0 V LB — V LA; and BS12 X ALB . "1 1" VLA3 v/LB3 And these after due reduction come forth __ SEC. XII.] OF NATURAL PHILOSOPHY. 2'3\. 2SI3 4 SI3 ~oj-p And these by subducting the last from the first, become -oT~r Therefore the entire force with ,7hich the corpuscle P is attracted towards the centre of the sphere is as-^, that is, reciprocally as PS3 X PJ Q.E.I. By the same method one may determine the attraction of a corpuscle situate within the sphere, but more expeditiously by the following Theorem. PROPOSITION LXXXIL THEOREM XLI. In a sphere described about the centre S with the interval SA, if there be taken SI, SA, SP continually proportional ; ! sat/, that the attraction, of a corpuscle within the sphere in any place I is to its attraction without the sphere in the place P in a ratio compounded of the subduplicate ratio of IS, PS, the distances from the centre, and the subduplicate ratio of tJie centripetal forces tending to the centre in those places P and I. As if the centripetal forces of the particles of the sphere be reciprocally ;is the distances of the corpuscle at tracted by them ; the force with which the corpuscle situate in I is attracted by the entire sphere will be to the force with which it is attracted in P in a ratio compounded of the subdu plicate ratio of the distance SI to the distance SP, and the subduplicate ratio of the centripetal force in the place I arising from any particle in the centre to the centripetal force in the place P arising from the same particle in the centre ; that is, in the subduplicate ratio of the distances SI, SP to each other reciprocally. These two subduplicate ratios compose the ratio of equality, and therefore the attractions in I and P produced by the whole sphere are equal. By the like calculation, if the forces of the particles of the sphere are reciprocally in a duplicate ratio of the distances, it will be found that the attraction in I is to the attraction in P as the distance SP to the semi -diameter SA of the sphere. If those forces are reciprocally in a triplicate ratio of the distances, the attractions in I and P will be to each other as SP2 to SA3 ; if in a quadruplicate ratio, as SP3 to SA3. There fore since the attraction in P was found in this last case to be reciprocally as PS 3 X PI, the attraction in I will be reciprocally as S A 3 X PI, that is, because S A 3 is given reciprocally as PI. And the progression is the same in injinitnm. The demonstration of this Theorem is as follows : The things remaining as above constructed, and a corpuscle being in anj 332 THE MATHEMATICAL PRINCIPLES [BOOK I. place P. the ordinate DN was found to be as T)F2 \" PS — 00 ^~\r- Therefore if I Cj X V IE be drawn, that ordinate for any other place of the corpuscle, as I, will DE2 X IS become (mutatis mutandis] as ~T~p~rry~- Suppose the centripetalsorces flowing from any point of the sphere, as E, to be to each other at the dis tances IE and PE as PE'1 to IE11 (where the number u denotes the index DE2 X PS of the powers of PE and IE), and those ordinates will become as ^p - -57^7, 2 \x IS and — ~" --- TT7, whose ratio to each other is as PS X IE X IEn to IS X IE X IE" PE X PEn. Because SI, SE, SP are in continued proportion, the tri angles SPE, SEI are alike ; and thence IE is to PE as IS to SE or SA. For the ratio of IE to PE write the ratio of IS to SA ; and the ratio of the ordinates becomes that of PS X IE" to SA X PEn. But the ratio of PS to SA is snbduplicate of that of the distances PS, SI ; and the ratio of IE" to PE1 (because IE is to PE as IS to SA) is subduplicate of that of the forces at the distances PS, IS. Therefore the ordinates, and conse quently the areas whioifi the ordinates describe, and the attractions propor tional to them, are in a ratio compounded of those subduplicate ratios. Q.E.D. PROPOSITION LXXXIII. PROBLEM XLII. To find the force with which a corpuscle placed in the centre of a sphere is attracted towards any segment of that sphere whatsoever. Let P be a body in the centre of that sphere and RBSD a segment thereof contained under the plane RDS, and thesphrcrical superficies RBS. Let DB be cut in F by a sphaerical superficies EFG described from the centre P, and let the segment be divided into the parts _B BREFGS, FEDG. Let us suppose that segment to be not a purely mathematical but a physical superficies, having some, but a perfectly inconsiderable thickness. * Let that thickness be called O, and (by what Archi medes has demonstrated) that superficies will be as PF X DF X O. Let us suppose besides the attrac tive forces of the particles of the sphere to be reciprocally as that power of r.he distances, of which n is index ; and the force with which the superficies DE2 X O EFG attracts the body P will be (by Prop. LXXIX) as -- — that, 2DF X O is, as ---?— -,- DF2 X O ~"~ppn — * ppn the perpendicular FN drawn into SEC. XJ11.I OF NATURAL PHILOSOPHY. 233 O be proportional to this quantity ; and the curvilinear area BDI, which the ordinate FN, drawn through the length DB with a continued motion will describe, will be as the whole force with which the whole segment RBSD attracts the body P. Q.E.I. PROPOSITION LXXXIV. PROBLEM XLIII. To find the force with which a corpuscle, placed without the centre of a sphere iti the axis of any segment, is attracted by that segment. Let the body P placed in. the axis ADB of the segment KBK be attracted by that seg ment. About the centre P, with the interval PE, let the spherical superficies EFK be de- scribed; and let it divide the segment into two parts EBKFE and EFKDE. Find the force of the first of those parts by Prop. LXXXI, and the force of the latter part by Prop. LXXXIII, and the sum of the forces will be the force of the whole segment EBKDE. Q.E.I. SCHOLIUM. The attractions of sphaerical bodies being now explained, it comes next in order to treat of the laws of attraction in other bodies consisting in like manner of attractive particles ; but to treat of them particularly is not neces sary to my design. It will be sufficient to subjoin some general proposi tions relating to the forces of such bodies, and the motions thence arising, because the knowledge of these will be of some little use in philosophical inquiries. SECTION XIII. Of the attractive forces of bodies which are not of a sphcerical figure. PROPOSITION LXXXV. THEOREM XLIL If a body be attracted by another, and its attraction be vastly stronger when it is contiguous to the attracting body than when they are sepa rated from one another by a very small interval ; the forces of the particles of the attracting body decrease, in the recess of the body at tracted, in more than a duplicate ratio of the distance of the particles. For if the forces decrease in a duplicate ratio of the distances from the particles, the attraction towards a sphaerical body being (by Prop. LXXIV) reciprocally as the square of the distance of the attracted body from the centre of the sphere, will not be sensibly increased by the contact, and it 234 THE MATHEMATICAL PRINCIPLES [BOOK 1 \vill be still less increased by it, if the attraction, in the recess of the body attracted, decreases in a still less proportion. The proposition, therefore, is evident concerning attractive spheres. And the case is the same of con cave sphaerical orbs attracting external bodies. And much more does it appear in orbs that attract bodies placed within them, because there the attractions diffused through the cavities of those orbs are (by Prop. LXX) destroyed by contrary attractions, and therefore have no effect even in the place of contact. Now if from these spheres and sphoerical orbs we take away any parts remote from the place of contact, and add new parts any where at pleas ore, we may change the figures of the attractive bodies at pleasure ; but the parts added or taken away, being remote from the place of contact, will cause no remarkable excess of the attraction arising from the contact of the two bodies. '1 herefore the proposition holds good in bodies of all figures. Q.E.I). PROPOSITION LXXXV1. THEOREM XLIII. If the forces of the particles of which an attractive body is composed de crease, in. the recess of the attractive body, in a triplicate or more than a triplicate ratio of the distance from the particles, the attraction will be vastly stronger in the point of contact than when the attracting and attracted bodies are separated from each other, though by never so small an interval. For that the attraction is infinitely increased when the attracted corpus cle comes to touch an attracting sphere of this kind, appears, by the solu tion of Problem XLI, exhibited in ££e second and third Examples. The same will also appear (by comparing those Examples and Theorem XLI together) of attractions of bodies made towards concavo-convex orbs, whether the attracted bodies be placed without the orbs, or in the cavities within them. And by aiding to or taking from those spheres and orbs any at tractive matter any where without the place of contact, so that the attrac tive bodies may receive any assigned figure, the Proposition will hold good of all bodies universally. Q.E.D. PROPOSITION LXXXVII. THEOREM XI. IV. If two bodies similar to each other, and consisting of matter equally at tractive attract separately two corpuscles proportional to those bodies, and in a like situation to them, the accelerative attractions of the cor puscles towards the entire bodies will be as the acccleratire at tractions of the corpuscles towards particles of the bodies proportional to the wholes, and alike situated in them. For if the bodies are divided into particles proportional to the wholes, and alike situated in them, it will be, as the attraction towards any parti cle of one of the bodies to the attraction towards the correspondent particle SEC. A III.] OF NATURAL PHILOSOPHY. 235 in the other body, so are the attractions towards the several particles of the iirst body, to the attractions towards the several correspondent particles of the other body ; and, by composition, so is the attraction towards the first whole body to the attraction towards the second whole body. Q,.E.U. COR. 1 . Therefore if, as the distances of the corpuscles attracted increase, the attractive forces of the particles decrease in the ratio of any power of the distances, the accelerative attractions towards the whole bodies will be as the bodies directly, and those powers of the distances inversely. A* if the forces of the particles decrease in a duplicate ratio of the distances from the corpuscles attracted, and the bodies are as A 3 and B 3, and there fore both the cubic sides of the bodies, and the distance of the attracted corpuscles from the bodies, are as A and B ; the accelerative attractions A3 B3 towards the bodies will be as — and — , that is, as A and B the cubic sjides of those bodies. If the forces of the particles decrease in a triplicate ratio of the distances from the attracted corpuscles, the accelerative attrac- A3 B3 tions towards the whole bodies will be as — and 5--, that is, equal. If the A. tj forces decrease in a quadruplicate ratio, the attractions towards the bodies A3 B3 will be as-— an^ 04' *^at is, reciprocally as the cubic sides A and B. And so in other cases. COR. 2. Hence, on the other hand, from the forces with which like bodies attract corpuscles similarly situated, may be collected the ratio of the de crease of the attractive forces of the particles as the attracted corpuscle recedes from them ; if so be that decrease is directly or inversely in any ratio of the distances. PROPOSITION LXXXVIII. THEOREM XLV. If the attractive forces of the equal particles of any body be as the dis tance of the places from the particles, the force of the whole body will tend to its centre of gravity ; and will be the same with the force of a globe, consisting of similar and equal matter, and having its centre in the centre of gravity. Let the particles A, B, of the body RSTV at tract any corpuscle Z with forces which, suppos-| ing the particles to be equal between themselves, are as the distances AZ, BZ ; but, if they are supposed unequal, are as those particles and their distances AZ, BZ, conjunctly, or (if I may go speak) as those particles drawn into their dis tances AZ, BZ respectively. And let those forces be expressed by the 236 THE MATHEMATICAL PRINCIPLES [BOOK 1. contents u.ider A X AZ, and B X BZ. Join AB, and let it be cut in G, so that AG may be to BG as the particle B to the particle A : and G will be the common centre of gravity of the particles A and B. The force A X AZ will (by Cor. 2, of the Laws) be resolved into the forces A X GZ and A X AG ; and the force B X BZ into the forces B X GZ and B X BG. Now the forces A X AG and B X BG, because A is proportional to B, and BG to AG, are equal, and therefore having contrary directions de stroy one another. There remain then the forces A X GZ and B X GZ. These tend from Z towards the centre G, and compose the force A + B X GZ ; that is, the same force as if the attractive particles A and B were placed in their common centre of gravity G, composing there a little globe. By the same reasoning, if there be added a third particle G, and the force of it be compounded with the force A -f B X GZ tending to the cen tre G, the force thence arising will tend to the common centre of gravity of that globe in G and of the particle C ; that is, to the common centre oi gravity of the three particles A, B, C ; and will be the same as if that globe and the particle C were placed in that common centre composing a greater globe there ; and so we may go on in injinitum. Therefore the whole force of all the particles of any body whatever RSTV is the same as if that body, without removing its centre of gravity, were to put on the form of a globe. Q,.E.D. COR. Hence the motion of the attracted body Z will be the same as if the attracting body RSTV were sphaerical ; and therefore if that attract ing body be either at rest, or proceed uniformly in a right line, the body attracted will move in an ellipsis having its centre in the centre of gravity of the attracting body. PROPOSITION LXXXIX. THEOREM XLVI. If there be several bodies consisting of equal particles whose jorces are as the distances of the places from each, the force compounded of all the forces by which any corpuscle is attracted will tend to the common centre of gravity of the attracting bodies ; and will be the same as if those attracting bodies, preserving their common centre of gravity, should unite there, and be formed into a globe. This is demonstrated after the same manner as the foregoing Proposi tion. COR. Therefore the motion of the attracted body will be the same as if the attracting bodies, preserving their common centre of gravity, should unite there, and be formed into a globe. And, therefore, if the common centre of gravity of the attracting bodies be either at rest, or proceed uni formly in a right line, the attracted body will move in an ellipsis having Us centre in the common centre of gravity of the attracting bodies. SEC. XlII.j OF NATURAL PHILOSOPHY. 237 PROPOSITION XC. PROBLEM XLIV. If to the several points of any circle there tend equal centripeta forces, increasing' or decreasing in any ratio of the distances ; it is required to Jin d the force with which a corpuscle is attracted, that is, situate any where in a right line which stands at right angles to the plant of the circle at its centre. Suppose a circle to be described about the cen tre A with any interval AD in a plane to which ; the right line AP is perpendicular ; and let it be required to find the force with which a corpuscle P is attracted towards the same. From any point E of the circle, to the attracted corpuscle P, let there be drawn the right line PE. In the right line PA take PF equal to PE, and make a per- pendicular FK, erected at F, to be as the force with which the point E attracts the corpuscle P. And let the curve line IKL be the locus of the point K. Let that cu/, fe meet the plane of the circle in L. In PA take PH equal to PD, and p/^ct the perpendicular HI meeting that curve in I ; and the attraction of the corpuscle P towards the circle will be as the area AHIL drawn into the altitude AP Q.E.I. For let there be taken in AE a very small line Ee. Join Pe, and in PE, PA take PC, Pf equal to Pe. And because the force, with which any point E of the annulus described about the centre A with the interval AS in the aforesaid plane attracts to itself the body P, is supposed to be as FK ; and, therefore, the force with which that point attracts the body P AP X FK towards A is as - — ^p ; and the force with which the whole annulus AP X FK attracts tne body P towards A is as the annulus and p^ conjunct- ly ; and that annulus also is as the rectangle under the radius AE aad the breadth Ee, and this rectangle (because PE and AE, Ee and CE are pro portional) is equal to the rectangle PE X CE or PE X F/; the force *-ith which that annulus attracts the body P towards A will be as PE X AP X FK Ff and pp~~~ conjunctly ; that is, as the content under F/ X FK X AP, or as the area FKkf drawn into AP. And therefore the sum of the forces with which all the annuli, in the circle described about the centre A with the interval AD, attract the body P towards A, is as the whole area AHIKL drawn into AP. Q.E.D. COR. 1. Hence if the forces of the points decrease in the duplicate ratio 238 THE MATHEMATICAL PRINCIPLES [BOOK I of the distances, that is, if FK be as rfFK, and therefore the area AHIKL as p-7 — p- ; the attraction of the corpuscle P towards the circle will PA AH be as 1 — ; that is, as COR. 2. And universally if the forces of the points at the distances D b( reciprocally as any power Dn of the distances; that is, if FK be as — . and therefore the area AHIKL as 1 1 " — l PH" — 1 PA , ; the attraction of the corpuscle P towards the circle will be as PA" — 2 PH" — l COR. 3. And if the diameter of the circle be increased in itifinitum, and the number n be greater than unity ; the attraction of the corpuscle P to wards the whole infinite plane will be reciprocally as PA" — 2, because the PA other term vanishes. PROPOSITION XCI. PROBLEM XLV. To find the attraction of a corpuscle situate in the axis of a round solid, to whose several points there tend equal centripetal forces decreasing in any ratio of the distances whatsoever. Let the corpuscle P, situate in the axis AB of the solid DECG, be attracted towards that solid. Let the solid be cut by any circle as RFS, perpendicular to the axis ; and in its semi-diameter FS, in any plane PALKB pass ing through the axis, let there be taken (by Prop. XC) the length FK proportional to the force with which the corpuscle P is attracted towards that circle. Let the locus of the point K be the curve line LKI, meeting the planes of the outermost circles AL and BI in L and I ; and the attraction of the corpuscle P towards the solid will be as the area LABI. Q..E.I. COR. 1. Hence if the solid be a cylinder described by the parallelogram A DEB revolved about the axis AB, and the centripetal forces tending to the several points be reciprocally as the squares of the distances from the points ; the attraction of the corpuscle P towards this cylinder will be as AB — PE + PD. For the ordinate FK (by Cor. 1, Prop. XC) will be PF as 1 — --. The part 1 of this quantity, drawn into the length AB, de- SEC. XIII. OF NATURAL PHILOSOPHY 239 scribes the area 1 X AB ; and the other part PF , drawn into the length PB describes the ix area 1 into PE — AD (as may be easily shewn from the quadrature of the curve LKI); and, in like manner, the same part drawn into the length PA describes the area L into PD — AD. and drawn into AB, the "At G Iv S 13 M 7J" 1 difference of PB and PA, describes 1 into PE — PD, the difference of the areas. From the first content 1 X AB take away the last content 1 into PE — PD, and there will remain the area LABI equal to 1 into AB — PE -h PD. Therefore the force, being proportional to this area, is as AB — PE + PD. COR. 2. Hence also is known the force by which a spheroid AGBC attracts any body P situate externally in its axis AB. Let NKRM be a conic section whose or- dinate KR perpendicular to PE may be \ always equal to the length of the line PD, continually drawn to tlie point D in which that ordinate cuts the spheroid. From the vertices A, B, of the spheriod, let there be erected to its axis AB the perpendiculars AK, BM, respectively equal to AP. BP, and therefore meeting the conic section in K and M; and join KM cutting off from it the segment KMRK. Let S be the centre of the spheroid, and SC its greatest semi-diameter : and the force with which the spheroid attracts the body P will be to the force with which a sphere describ- , ....,,. ASxCS2-PSxKMRK ed with the diameter AhJ attracts the same body as prrr ^ r-= 1 o -f- Go2 — Ao AS3 is to fkT^,. And by a calculation founded on the same principles may be found the forces of the segments of the spheroid. COR. 3. If the corpuscle be placed within the spheroid and in its axis, the attraction will be as its distance from the centre. This may be easily collected from the following reasoning, whether the particle be in the axis or in any other given diameter. Let AGOF be an attracting sphe roid, S its centre, and P the body attracted. Through the body P let there be drawn the semi-diameter SPA, and two right lines DE, FC meeting the spheroid in 1) and E, F and G ; and let PCM, HLN be the superficies of 240 THE MATHEMATICAL PRINCIPLE* ffioOK 1. two interior spheroids similar and concentrical to the exterior, the first of which passes through the body P. and cuts the right lines DE, FG in B and C ; arid the latter cuts the same right lines in H and I, K and L. I ,et the spheroids have all one common axis, and the parts of the right lines intercepted on both sides DP and BE, FP and CG, DH and IE, FK and LG, will be mutually equal; because the right lines DE. PB, and HI. are bisected in the same point, as are also the right lines FG, PC, and KL. Conceive now DPF. EPG to represent opposite cones described with the infmitely small vertical angles DPF, EPG, and the lines DH, El to be infinitely small also. Then the particles of the cones DHKF, GLIE, cut off by the spheroidical superficies, by reason of the equality of the lines DH and ET; will be to one another as the squares of the distances from the body P, and will therefore attract that corpuscle equally. And by a like rea soning if the spaces DPF, EGCB be divided into particles by the superfi cies of innumerable similar spheroids concentric to the former and having J. O one common axis, all these particles will equally attract on both sides the body P towards contrary parts. Therefore the forces of the cone DPF. and of the conic segment EGCB, are equal, and by their contrariety de stroy each other. And the case is the same of the forces of all the matter that lies without the interior spheroid PCBM. Therefore the body P is attracted by the interior spheroid PCBM alone, and therefore (by Cor. 3, Prop. 1 , XXII) its attraction is to the force with which the body A is at tracted by the whole spheroid AGOD as the distance PS to the distance AS. Q.E.D. PROPOSITION XCII. PROBLEM XLVI. An attracting body being given, it is required to find the ratio of the de crease of the centripetal forces tending to its several points. The body given must be formed into a sphere, a cylinder, or some regu lar figure, whose law of attraction answering to any ratio of decrease may be found by Prop. LXXX, LXXXI, and XCI. Then, by experiments, the force of the attractions must be found at several distances, and the law of attraction towards the whole, made known by that means, will give the ratio of the decrease of the forces of the several parts ; which was to be found. PROPOSITION XCIII. THEOREM XLVII. If a solid be plane on one side, and infinitely extended on all otljer sides, and consist of equal particles equally attractive, iv hose forces decrease, in the recess from the solid, in the ratio of any power greater than the square of the distances ; and a corpuscle placed towards eit far part of the plane is attracted by the force of the whole solid ; I say that tfie attractive force of the whole solid, in the recess from its platw superfi- XIILj OF NATURAL PHILOSOPHY". 241 n H m G ties, will decrease in the ratio of a power whose side is the distance oj the corpuscle from the plane, and its index less by 3 than the index oj the power of the distances. CASE 1. Let LG/be the plane by which the solid is terminated. Let the solid lie on that hand of the plane that is to wards I, and let it be resolved into in- _. numerable planes mHM, //IN, oKO, (fee., parallel to GL. And first let the attracted body C be placed without the solid. Let there be drawn CGHI per pendicular to those innumerable planes, and let the attractive forces of the points of the solid decrease in the ratio of a power of the distances whose index is the number n not less than 3. Therefore (by Cor. 3, Prop. XC) the force with which any plane mHM attracts the point C is reciprocally as CHn— 2. In the plane mHM take the length HM reciprocally proportional to CH1— 2, and that force will be as HM. In like manner in the several planes /GL, //,TN, oKO, (fee., take the lengths GL, IN, KO, (fee., reciprocally proportional to CGn— 2, CI1— 2, CKn — 2, (fee., and the forces of those planes will bs as the lengths so taken, and therefore the sum of the forces as the sum of the lengths, that is, the force of the Avhole solid as the area GLOK produced infinitely towards OK. But that area (by the known methods of quadratures) is reciprocally as CGn— 3, and therefore the force of the whole solid is reciprocally as CG"-3. Q.E.D. CASE 2. Let ttecorpuscleC be now placed on that hand of the plane /GL that is within the solid, and take the distance CK equal to the distance CG. And the part of the solid LG/oKO termi nated by the parallel planes /GL, oKO, will at tract the corpuscle C, situate in the middle, neither one way nor another, the contrary actions of the opposite points destroying one another by reason of their equality. Therefore the corpuscle C is attracted by the force only of the solid situate beyond the plane OK. But this force (by Case 1) is reciprocally as CKn—3, that is, (because CG, CK are equal) reciprocally as CG"— 3. Q,.E.D. COR. 1. Hence if the solid LGIN be terminated on each sitfe by two in finite parallel places LG, IN, its attractive force is known, subducting from the attractive force of the whole infinite solid LGKO the attractive force of the more distant part NIKO infinitely produced towards KO. COR. 2. If the more distant part of this solid be rejected, because its at traction compared with the attraction of the nearer part is inconsiderable, 16 0 N K 1 0 C 242 THE MATHEMATICAL PRINCIPLES [BOOK 1 the attraction of that nearer part will, as the distance increases, decrease nearly in the ratio of the power CGn— 3. Con. 3. And hence if any finite body, plane on one side, attract a cor puscle situate over against the middle of that plane, and the distance between the corpuscle and the plane compared with the dimensions of the attracting body be extremely small ; and the attracting body consist of homogeneous particles, whose attractive forces decrease in the ratio of any power of the distances greater than the quadruplicate ; the attractive force of the whole body will decrease very nearly in the ratio of a power whose side is that very small distance, and the index less by 3 than the index of the former power. This assertion does not hold good, however, of a body consisting of particles whose attractive forces decrease in the ratio of the triplicate power of the distances ; because, in that case, the attraction of the remoter part of the infinite body in the second Corollary is always infinitely greater than the attraction of the nearer part. SCHOLIUM. If a body is attracted perpendicularly towards a given plane, and from the law of attraction given, the motion of the body be required ; the Pro blem will be solved by seeking (by Prop. XXXIX) the motion of the body descending in a right line towards that plane, and (by Cor. 2, of the Laws) compounding that motion with an uniform motion performed in the direc tion of lines parallel to that plane. And, on the contrary, if there be re quired the law of the attraction tending towards the plane in perpendicu lar directions, by which the body may be caused to move in any given curve line, the Problem will be solved by working after the manner of the third Problem. But the operations may be contracted by resolving the ordinates into converging series. As if to a base A the length B be ordinately ap plied in any given angle, and that length be as any power of the base A^ ; and there be sought the force with which a body, either attracted to wards the base or driven from it in the direction of that ordinate, may be caused to move in the curre line which that ordinate always describes with its superior extremity ; I suppose the base to be increased by a very small ,m m part O, and I resolve the ordinate A -f Ol^ into an infinite series A- -f !!L OA ^ + '^-^--- OOA — ;- &c., and I suppose the force proper- 11 £1111 tional to the term of this series in which O is of two dimensions, that is, to the term - - OOA ^YT, Therefore the force sought is aa SEC. XIV.J OF NATURAL PHILOSOPHY. 2M mm — mn m — 2n .... mm — mn m — 2n A — 7, — , or, which is the same thinor, as L> m . nn nn As if the ordinate describe a parabola, m being — 2, and n = 1, the force will be as the given quantity 2B°, and therefore is given. Therefore with a given force the body will move in a parabola, as Galileo has demon strated. If the ordinate describe an hyperbola, m being = 0 — 1, and n — 1, the force will be as 2 A 3 or 2B 3 ; and therefore a force which is as the cube of the ordinate will cause the body to move in an hyperbola. But leaving this kind of propositions, I shall go on to some others relating to motion which I have fiot yet touched upon. SECTION XIV. Of the motion of very small bodies when agitated by centripetal forces tending to the several parts of any very great body. PROPOSITION XCIV. THEOREM XLVIII. If two similar mediums be separated from each other by a space termi nated on both sides by parallel planes, and a body in its passage through that space be attracted or impelled perpendicularly towards either of those mediums, and not agitated or hindered by any other force ; and the attraction be every where the same at equal distances from either plane, taken towards the same hand of the plane ; I say, that the sine of incidence upon either plane will be to the sine of emcr gence from the other plane in a given ratio. CASE 1. Let Aa and B6 be two parallel planes, and let the body light upon the first plane Aa in the direction of the line GH, and in its whole passage through the intermediate space let it be attracted or impelled towards the medium of in cidence, and by that action let it be made to de scribe a curve line HI, and let it emerge in the di rection of the line IK. Let there be erected IM perpendicular to Eb the plane of emergence, and meeting the line of incidence GH prolonged in M, and the plane of inci dence Aa in R ; and let the line of emergence KI be produced and meet HM in L. About the centre L, with the interval LI, let a circle be de scribed cutting both HM in P and Q, and MI produced in N ; and, first, if the attraction or impulse be supposed uniform, the curve HI (by what Galileo has demonstrated) be a parabola, whose property is that of a roc- THE MATHEMATICAL PRINCIPLES [BoOK 1 tangle under its given latiis rectum and the line IM is equal to the squarrf cf HM ; and moreover the line HM will be bisected in L. Whence if to MI there be let fall the perpendicular LO, MO, OR will be equal; and adding the equal lines ON, OI, the wholes MN, IR will be equal also. Therefore since IR is given, MN is also given, and the rectangle NMI is to the rectangle under the latus rectum and IM, that is, to HMa in a given ratio. But the rectangle NMI is equal to the rectangle PMQ,, that is, to the difference of the squares ML2, and PL2 or LI2 ; and HM2 hath a given ratio to its fourth part ML2; therefore the ratio of ML2 — LI2 to ML2 is given, and by conversion the ratio of LI2 to ML ', and its subduplicate, theratrio of LI to ML. But in every triangle, as LMI, the sines jf the angles are proportional to the opposite sides. Therefore the ratio of the sine of the angle of incidence LMR to the sine of the angle of emergence LIR is given. QJE.lr). CASE 2. Let now the body pass successively through several spaces ter minated with parallel planes Aa/>B, B6cC, &c., and let it be acted on by a \ . force which is uniform in each of them separ- \ a ately, but different in the different spaces ; and B \ fr by what was just demonstrated, the sine of the c ^^ c angle of incidence on the first plane Aa is to the sine of emergence from the second plane Bb in a given ratio ; and this sine of incidence upon the second plane Bb will be to the sine of emergence from the third plane Cc in a given ratio ; and this sine to the sine of emergence from the fourth plane Dd in a given ra tio ; and so on in infinitum ; and, by equality, the sine of incidence on the first plane to the sine of emergence from the last plane in a given ratio. I ,et now the intervals of the planes be diminished, and their number be in finitely increased, so that the action of attraction or impulse, exerted accord ing to any assigned law, may become continual, and the ratio of the sine of incidence on the first plane to the sine of emergence from the last plane being all along given, will be given then also. QJE.D. PROPOSITION XCV. THEOREM XLIX. The same thing's being supposed, I say, that the velocity of the body be fore its incidence is to its velocity after emergence as the sine of emer gence to the sine of incid nee. Make AH and Id equal, and erect the perpen- diculars AG, dK meeting the lines of incidence and emergence GH, IK, in G and K. In GH --« take TH equal to IK, and to the plane Aa let ^ fall a perpendicular TV. And (by Cor. 2 of the |x^ I Laws of Motion) let the motion of the body be jv- resolved into two, one perpendicular to the planes SEC. X1V.J OF NATURAL PHILOSOPHY. 245 Aa, Bb, Cc, &c, and another parallel to them. The force of attraction or impulse, acting in directions perpendicular to those planes, does not at all alter the motion in parallel directions ; and therefore the body proceeding with this motion will in equal times go through those equal parallel inter vals that lie between the line AG and the point H, and between the point I and the line dK ; that is, they will describe the lines GH, IK in equal times. Therefore the velocity before incidence is to the velocity after emergence as GH to IK or TH, that is, as AH or Id to vH, that is (sup posing TH or IK radius), as the sine of emergence to the sine of inci dence. Q.E.D. PROPOSITION XCVL THEOREM L. The same things being supposed, and that the motion before incidence is swifter than afterwards ; 1 sat/, lhat if the line of incidence be in clined continually, the body will be at last reflected, and the angle of reflexion will be equal to the angle of incidence. For conceive the body passing between the parallel planes Aa, Bb, Cc, &c., to describe parabolic arcs as above; and let those arcs be HP, PQ, QR, &c. And let the obliquity of the line of inci- g dence GH to the first plane Aa be such rc~ £ that the sine of incidence may be to the radius of the circle whose sine it is, in the same ratio which the same sine of incidence hath to the sine of emer gence from the plane Dd into the space DefeE ; and because the sine of emergence is now become equal to radius, the angle of emergence will be a right one, and therefore the line of emergence will coincide with the plane Dd. Let the body come to this plane in the point R ; and because the line of emergence coincides with that plane, it is manifest that the body can proceed no farther towards the plane Ee. But neither can it proceed in the line of emergence Rd; because it is perpetually attracted or impelled towards the medium of incidence. It will return, therefore, between the planes Cc, Dd, describing an arc of a parabola Q,R</, whose principal vertex (by what Galileo has demonstrated) is in R, cutting the plane Or in the same angle at q, that it did before at Q, ; then going on in the parabolic arcs qp, ph, &c., similar and equal to the former arcs QP, PH, &c., it will cut the rest of the planes in the same angles at p, h, (fee., as it did before in P, H, (fee., and will emerge at last with the same obliquity at h with which it first impinged on that plane at H. Conceive now the intervals of the planes Aa, Bb, Cc, Dd, Ee, (fee., to be infinitely diminished, and the number in finitely increased, so that the action of attraction or impulse, exerted ac cording to any assigned law, may become continual; and, the angle of emergence remaining all alor g equal to the angle of incidence, will be equal to the same also at last. Q.E.D. 246 THE MATHEMATICAL PRINCIPLES IBoOK 1 SCHOLIUM. These attractions bear a great resemblance to the reflexions and refrac tions of light made in a given ratio of the secants, as was discovered h} Siiellius ; and consequently in a given ratio of the sines, as was exhibited by Hes Cortes. For it is now certain from the phenomena of Jupiter's ^satellites, confirmed by the observations of different astronomers, that light is propagated in succession, and requires about seven or eight minutes to travel from the sun to the earth. Moreover, the rays of light that are in our air (as lately was discovered by Grimaldus, by the admission of light into a dark room through a small hole, which 1 have also tried) in their passage near the angles of bodies, whether transparent or opaque (such aa the circular and rectangular edges of gold, silver and brass coins, or of knives, or broken pieces of stone or glass), are bent or inflected round those bodies as if they were attracted to them ; and those rays which in their passage come nearest to the bodies are the most inflected, as if they were most attracted : which tiling I myself have also carefully observed. And those which pass at greater distances are less inflected ; and those at still greater distances are a little inflected the contrary way, and form three fringes of colours. In the figure 5 represents the edge of a knife, or any tl \ / K", \ / ^•::^ x -^» . v ^--:<*, -f:::r;^c :.-/ N «>V J V U W~"~a~" "~a C: O la kind of wedge AsB : and gowog,fmnif,emtme, dlsld, are rays inflected to wards the knife in the arcs owo, nvn, mtm, Isl ; which inflection is greater or less according; to their distance from the knife. Now since this inflec tion of the rays is performed in the air without the knife, it follows that the rays which fall upon the knife are first inflected in the air before they touch the knife. And the case is the same of the rays falling upon glass. The refraction, therefore, is made not in the point of incidence, but gradually, by a continual inflection of the rays ; which is done partly in the air before they touch the glass, partly (if [ mistake not) within the glass, after they have entered it ; as is represented in the rays ckzc, bujb^ ahxa, falling upon r, q, p, and inflected between k and z, i and y, h and x. Therefore because of the analogy there is between the propagation of the rays f light and the motion of bodies, I thought it not amiss to add the followi »g Propositions far optical uses ; not at all. considering the nature of the rays of .light, or inquiring whether they are bodies or not ; but only determining the tra jectories of bodies which are extremely like the trajectories of the rays. SEC. XIV.] OF NATURAL PHILOSOPHY. 247 PROPOSITION XCVII. PROBT.-EM XL VII. Supposing t/w sine of incidence upon any superficies to be in a given ra tio to the sine of emergence ; and that tha inflection of t/ts paths of those bodies near that superficies is performed in a very short space, which may be considered as a point ; it is required to determine suck a superficies as may cause all the corpuscles issuing from any one given place to converge to another given place. Let A be the place from whence the cor- E puscles diverge ; B the place to which they should converge ; CDE the curve line which by its revolution round the axis AB describes . /C the superficies sought ; D, E, any two points of that curve ; and EF, EG, perpendiculars let fall on the paths of the bodies AD, DB. Let the point D approach to and coalesce with the point E ; and the ultimate ratio of the line DF by which AD is increased, to the line DG by which DB is diminished, will be the same as that of the sine of incidence to the sine of emergence Therefore the ratio of the increment of the line AD to the decrement of the line DB is given: and therefore if in the axis AB there be taken any where the point C through which the curve CDE must pass, and CM the increment of AC be taken in that given ratio to CN the decrement of BC, and from the centres A, B, with the intervals AM, BN, there be described two circles cutting each other in D ; that point D will touch the curve sought CDE, and, by touching it any where at pleasure, will determine that curve. Q.E.I. COR. 1. By causing the point A or B to go off sometimes in infinitum, and sometimes to move towards other parts of the point C, will bo obtain ed all those figures which Cartesins has exhibited in his Optics and Geom etry relating to refractions. The invention of which Cartcsius having thought fit to conceal, is here laid open in this Proposition. COR. 2. If a body lighting on any superfi cies CD in the direction of a ri^ht line AD, Qj- O \ » drawn according to any law, should emerge in the direction of another right line DK ; and from the point C there be drawn curve lines CP, CQ, always perpendicular to AD, DK ; the increments of the lines PD, QD, and therefore the lines themselves PD, Q.D, generated by those increments, will be as the sines of incidence and emergence to each other, and e contra. PROPOSITION XCVIII. PROBLEM XLVIII. The same things supposed ; if round the axis AB any attractive super ficies be described as CD, regular or irregular, through which the bo dies issuing from the given place A must pass ; it is required to find THE MATHEMATICAL PRINCIPLES. [BOOK J a second attractive superficies EF, which may make those bodies con verge to a given place B. Let a line joining AB cut the first superficies in C and the second in E, the point D being taken any how at plea sure. And supposing the f sine of incidence on the first superficies to the sine of emergence from the same, and the sine of emergence from the second super ficies to the sine of incidence on the same, to be as any given quantity M to another given quantity N; then produce AB to G, so that BG may be to CE as M — N to N ; and AD to H, so that AH may be equal to AG ; arid DF to K, so that DK may be to DH as N to M. Join KB, and about the centre D with the interval DH describe a circle meeting KB produced in L, and draw BF parallel to DL; and the point F will touch the line EF, which, being turned round the axis AB, will describe the superficies sought. Q.E.F. For conceive the lines CP, CQ to be every where perpendicular to AD, DF, and the lines ER, ES to FB, FD respectively, and therefore QS to be always equal to CK; and (by Cor. 2, Prop. XCVII) PD will be to QD as M to N, and therefore as DL to DK, or FB to FK ; and by division as DL — FB or PH — PD — FB to FD or FQ — QD ; arid by composition as PH— FB to FQ, that is (because PH and CG, QS and CE, are equal), as CE + BG — FR to CE — FS. But (because BG is to CE as M — N to N) it. comes to pass also that CE + BG is to CE as M to N; and therefore, by division, FR is to FS as M to N ; and therefore (by Cor. 2, Prop XCVI1) the superficies EF compels a body, falling upon it in the direction DF, to go on in the line FR to the place B. Q.E.D. SCHOLIUM. ,In the same manner one may go on to three or more superficies. But of all figures the sphserical is the most proper for optical uses. If the ob ject glasses of telescopes were made of two glasses of a sphaerical figure, containing water between them, it is not unlikely that the errors of the refractions made in the extreme parts of the superficies of the glasses may be accurately enough corrected by the refractions of the water. Such ob ject glasses are to be preferred before elliptic and hyperbolic glasses, not only because they may be formed with more ease and accuracy, but because the pencils of rays situate without the axis of the glass would be more accu rately refracted by them. But the different refrangibility of different raya is the real obstacle that hinders optics from being made perfect by sphaeri cal or any other figures. Unless the errors thence arising can be corrected, all the labour spent in correcting the others is quite thrown away. BOOK II BOOK II. OF THE MOTION OF BODIES. SECTION I. Of the motion of bodies that are resisted in the ratio of the velocity. PROPOSITION I. THEOREM I. Tf a body is resisted in the ratio of its velocity, the motion lost by re sistance is as the space gone over in its motion. For since the motion lost in each equal particle of time is as the velocity, that is, as the particle of space gone over, then, by composition, the motion lost in the whole time will he as the whole space gone over. Q.E.D. COR. Therefore if the body, destitute of all gravity, move by its innate force only in free spaces, and there be given both its whole motion at the beginning, and also the motion remaining after some part of the way is gone over, there will be given also the whole space which the body can de scribe in an infinite time. For that space will be to the space now de scribed as the whole motion at the beginning is to the part lost of that motion. LEMMA I. Quantities proportional to their differences are continually proportional. Let A be to A — B as B to B — C and C to C — D, (fee., and, by con version, A will be to B as B to C and C to D, &c. Q.E.D. PROPOSITION II. THEOREM II. If a body is resisted in the ratio of its velocity, and moves, by its vis in- sita only, through a similar medium, and the times be taken equal, the velocities in the beginning of each of the times are in a geometri cal progression, and the spaces described in each of the times are as the velocities. CASE 1. Let the time be divided into equal particles ; and if at the very beginning of each particle we suppose the resistance to act with one single impulse which is as the velocity, the decrement of the velocity in each of THE MATHEMATICAL PRINCIPLES [BOOK II. the particles of time will be as the same velocity. Therefore the veloci ties are proportional to their differences, and therefore (by Lem. 1, Book II) continually proportional. Therefore if out of an equal number of par ticles there be compounded any equal portions of time, the velocities at the beginning of those times will be as terms in a continued progression, which are taken by intervals, omitting every where an equal number of interme diate terms. But the ratios of these terms are compounded of the equaj ratios of the intermediate terms equally repeated, and therefore are equal Therefore the velocities, being proportional to those terms, are in geomet rical progression. Let those equal particles of time be diminished, and their number increased in infinitum, so that the impulse of resistance may become continual ; and the velocities at the beginnings of equal times, al ways continually proportional, will be also in this case continually pro portional. Q.E.D. CASE 2. And, by division, the differences of the velocities, that is, the parts of the velocities lost in each of the times, are as the wholes ; but the spaces described in each of the times are as the lost parts of the velocities (by Prop. 1, Book I), and therefore are also as the wholes. Q.E.D. TT COROL. Hence if to the rectangular asymptotes AC, CH, the hyperbola BG is described, and AB, DG be drawn per pendicular to the asymptote AC, and both the velocity of . the body, and the resistance of the medium, at the very be ginning of the motion, be expressed by any given line AC, and, after some time is elapsed, by the indefinite line DC ; the time may be expressed by the area ABGD, and the space described in that time by the line AD. For if that area, by the motion of the point D, be uniform ly increased in the same manner as the time, the right line DC will de crease in a geometrical ratio in the same manner as the velocity ; and the parts of the right line AC, described in equal times, will decrease in the same ratio. PROPOSITION III. PROBLEM I. To define the motion of a body which, in a similar medium, ascends or descends in a right line, and is resisted in the ratio of its velocity, and acted upon by an uniform force of gravity. The body ascending, let the gravity be expound ed by any given rectangle BACH ; and the resist ance of the medium, at the beginning of the ascent, by the rectangle BADE, taken on the contrary side Jfl e B^l | L- of the right line AB. Through the point B, with the rectangular asymptotes AC, CH, describe an hyperbola, cutting the perpendiculars DE, de, ID SEC. I.j OF NATURAL PHILOSOPHY. 253 G, g ; and the body ascending will in the time DGgd describe the space EG«-e; in the time DGBA, the space of the whole ascent EGB ; in the time ABK1, the space of descent BFK ; and in the time IKki the space of descent KFfk; and the velocities of the bodies (proportional to the re sistance of the medium) in these periods of time will be ABED, ABed, O, ABFI, AB/z respectively ; and the greatest velocity which the body can acquire by descending will be BACH. For let the rectangle BACH be resolved into in numerable rectangles AA', K/, Lm, M//, *fea, which shall be as the increments of the velocities produced in so many equal times; then will 0, AAr, AL Am, An, ifec., be as the whole velocities, and therefore (by suppo sition) as the resistances of the medium in the be- ginning of each of the equal times. Make AC to AJLLB AK, or ABHC to AB/vK, as the force of gravity to the resistance in the beginning of the second time ; then from the force of gravity subduct the resistances, and ABHC, K/vHC, L/HC, MwHC, (fee., will be as the abso lute forces with which the body is acted upon in the beginning of each of the times, and therefore (by Law I) as the increments of the velocities, that is, as the rectangles AA-, K/, Lm, M//, (fee., and therefore (by Lem. 1, Book II) in a geometrical progression. Therefore, if the right lines K£, LI M/TO, N//, &c., are produced so as to meet the hyperbola in q, r, s, t, (fee.. the areas AB^K, K</rL, LrsM, MsJN, (fee., will be equal, and there fore analogous to the equal times and equal gravitating forces. But the area AB^K (by Corol. 3, Lem. VII and VIII, Book I) is to the area Bkq as K^ to \kq, or AC to |AK, that is, as the force of gravity to the resist ance in the middle of the first time. And by the like reasoning, the areas <?KLr, rLMs, sMN/, (fee., are to the areas qklr, rims, smnt, (fee., as the gravitating forces to the resistances in the middle of the second, third, fourth time, and so on. Therefore since the equal areas BAKq, </KLr, rLMs, sMN/, (fee., are analogous to the gravitating forces, the areas Bkq, qklr, rims, smut, (fee., will be analogous to the resistances in the middle of each of the times, that is (by supposition), to the velocities, and so to the spaces described. Take the sums of the analogous quantities, and the areas Bkq, Elr, Ems, But, (fee., will be analogous to the whole spaces described ; and also the areas AB<?K, ABrL, ABsM, AB^N, (fee., to the times. There fore the body, in descending, will in any time ABrL describe the space Blr, and in the time Lr^N the space rlnt. Q,.E.D. And the like demonstra tion holds in ascending motion. COROL. 1. Therefore the greatest velocity that the body can acquire by falling is to the velocity acquired in any given time as the £iven force ol gravity which perpetually acts upon it to the resisting force which opposes it at the end of that time. 854 THE MATHEMATICAL PRINCIPLES [BOOK IL COROL. 2. But the time being augmented in an arithmetical progression, the sum of that greatest velocity and the velocity in the ascent, and also their difference in the descent, decreases in a geometrical progression. COROL. 3. Also the differences of the spaces, which are described in equal differences of the times, decrease in the same geometrical progression. COROL. 4. The space described by the body is the difference of two spaces, whereof one is as the time taken from the beginning of the descent, and the other as the velocity; which [spaces] also at the beginning of the descent are equal among themselves. PROPOSITION IV. PROBLEM II. Supposing the force of gravity in any similar medium to be uniform, and to tend perpendicularly to the plane of the horizon ; to define the motion of a projectile therein, which suffers resistance proportional to its velocity. Let the projectile go from any place D in the direction of any right line DP, and let its velocity at the beginning of the motion be expounded by the length DP. From the point P let fall the perpendicular PC on the horizontal line DC, and cut DC in A, so that DA may be to AC as the resistance of the medium arising from the motion up wards at the beginning to the force of grav ity; or (which comes to the same) so that t ie rectangle under DA and DP may be to that under AC and CP as the whole resist ance at the beginning of the motion to the force of gravity. With the asymptotes DC, CP describe any hyperbola GTBS cut ting the perpendiculars DG, AB in G and B ; complete the parallelogram DGKC, and let its side GK cut AB in Q,. Take N in the same ratio to QB as DC is in to CP ; and from any point R of the right line DC erect RT perpendicular to it, meeting the hy] erbola'in T, and the right lines EH, GK, DP in I, t, and V ; in that perpendicular take Vr equal to — ~- , or which is the same thing, take Rr equal to (""""PIT? — ^T— ; and the projectile in the time DRTG will arrive at the point r describing the curve line DraF, the locus of the point r ; thence it will come to its greatest height a in the perpendicular AB j and afterwards SEC. 1.J » OF NATURAL PHILOSOPHY. 255 ever approach to the asymptote PC. And its velocity in any pjint r will be as the tangent rL to the curve. Q.E.I. For N is to Q,B as DC to CP or DR to RV, and therefore RV is equal to PR X QB ,' -•..". "v v DRXQB-/GT — ^r - , and R/' (that is, RV — Vr, or - -- ^ --- ) is equal to DR X Ap — RDGT -- ~ --- . JNow let the time be expounded by the area RDGT and (by Laws, Cor. 2), distinguish the motion of the body into two others, one of ascent, the other lateral. And since the resistance is as the motion, let that also be distinguished into two parts proportional and contrary to the parts of the motion : and therefore the length described by the lateral motion will be (by Prop. II, Book II) as the line DR, and the height (by Prop. Ill, Book II) as the area DR X AB — RDGT, that is, as the line Rr. But in the very beginning of the motion the area RDGT is equal to the rectangle DR X AQ, and therefore that line Rr (or DRx AB — that is, as CP to DC ; and therefore as the motion upwards to the motion lengthwise at the beginning. Since, therefore, Rr is always as the height, and DR always as the length, and Rr is to DR at the beginning as the height to the length, it follows, that Rr is always to DR as the height to the length ; and therefore that the body will move in the line DraF. which is the locus of the point r. QJE.D. DR X AB RDGT COR. 1. Therefore Rr is equal to -- ^ ------ ^ - . and therefore if RT be produced to X so that RX may be equal to -- ^ -- ; that is, if the parallelogram ACPY be completed, and DY cutting CP in Z be drawn, and RT be produced till it meets DY in X ; Xr will be equal to RDGT — ^ — , and therefore proportional to the time. COR. 2. Whence if innumerable lines CR, or, which is the same, innu merable lines ZX, be taken in a geometrical progression, there will be as many lines Xr in an arithmetical progression. And hence the curve DraF is easily delineated by the table of logarithms. COR. 3. If a parabola be constructed to the vertex D, and the diameter DG produced downwards, and its latus rectum is to 2 DP as the whole resistance at the beginning of the notion to the gravitating force, the ve locity with which the body ought *o go from the place D, in the direction of the right line DP, so as in an uniform resisting medium to describe the curve DraF, will be the same as that with which it ought to go from the same place D in the direction of the same right line DP, so as to describe 256 THE MATHEMATICAL PRINCIPLES ~ [BOOK II I a parabola in a non-resisting medium. For the latus rectum of this parabola, at the very DV2 beginning of the motion, is -y-', andVris tGT DR x T* -~JT- or £^T — . But a right line, which, if drawn, would touch the hyperbola GTS in K G, is parallel to DK, and therefore T* is CKX DR c QBx DC ^ , and N is — ~pp • Ahd there- DC DR2 X CK x CP fore Vr is equal to 2DC2 X QlT~; *^at *S (Because D^ an<* ^)C, DV DV2 x CK ~x OP and DP are proportionals), to — ^T5 Fcrr — J an<* tne ^atus reeturn DV2 - comes out - 2DP2 X QB are proportional), CK X CP 2DP2 X DA AC X CP CP X AC ; that is, as the resistance to the gravity. (because and therefore is to 2DP as DP X DA to Q.E.D. COR. 4. Hence if a body be projected from any place D with a given velocity, in the direction of a right line DP given by posi tion, and the resistance of the medium, at the beginning of the motion, be given, the curve DraF, which that body will describe, may be found. For the velocity being given, the latus rectum of the parabola is given, as is well known. And taking 2DP to that latus rectum, as the force of gravity to the resisting force, DP is also given. Then cutting DC in A, so that GP X AC may be to DP X DA in the same ratio of the gravity to the resistance, the point A will be given. And hence the curve DraF is also given. COR. 5. And, on the contrary, if the curve DraF be given, there will be given x>th the velocity of the body and the resistance of the medium in each of the places r. For the ratio of CP X AC to DP X DA being given, there is given both the resistance of the medium at the beginning of the motion and the latus rectum of the parabola ; and thence the velocity at the be ginning of the motion is given also. Then from the length of the tangent SEC. I.] OF NATURAL PHILOSOPHY. 257 L there is given both the velocity proportional to it, and the resistance proportional to the velocity in any place r. COR. 6. But since the length 2DP is to the latus rectum of the para bola as the gravity to the resistance in D ; and, from the velocity aug mented, the resistance is 'ti gmented in the same ratio, but the latus rectum of the parabola is augmented in the duplicate of that ratio, it is plain thot the length 2DP is augmented in that simple ratio only ; and is therefore always proportional to the velocity ; nor will it be augmented or dimin ished by the change of the angle CDP, unless the velocity be also changed. COR. 7. Hence appears the method of deter mining the curve DraF nearly from the phe- nomena, and thence collecting the resistance and velocity with which the body is projected. Let two similar and equal bodies be projected with the same velocity, from the place D, in differ ent angles CDP, CDp ; and let the places F, f. where they fall upon the horizontal plane DC, be known. Then taking any length for D */ F DP or Dp suppose the resistance in D to be to the suavity in any ratio whatsoever, and let that ratio be expounded by any length SM. Then, , _ by computation, from that assumed length DP, ^x find the lengths DF, D/; and from the ratio F/ -p^, found by calculation, subduct the same ratio as found by experiment ; and let the cKfference be expounded by the perpendicular MN. Repeat the same a second and a third time, by assuming always a new ratio SM of the resistance to the gravity, and collecting a new difference MN. Draw the affirmative differences on one side of the right line SM, and the negative on the other side ; and through the points N, N, N, draw a regular curve NNN. cutting the right line SMMM in X, and SX will be the true ratio of the resistance to the gravity, which was to be found. From this ratio the length DF is to be collected by calculation ; and a length, which is to the assumed length DP as the length DF known by experiment to the length DF just now found, will be the true length DP. This being known, you will have both the curve line DraF which the body describes, and also the velocity and resistance of the body in each place. SCHOLIUM. But, yet, that the resistance of bodies is in the ratio of the velocity, is more a mathematical hypothesis than a physical one. In mediums void of all te nacity, the resistances made to bodies are in the duplicate ratio of the ve locities. For by the action of a swifter body, a greater motion in propor- 17 THE MATHEMATICAL PRINCIPLES [BoOK IL tion to a greater velocity is communicated to the same quantity of the medium in a less time ; and in an equal time, by reason of a greater quan tity of the disturbed medium, a motion is communicated in the duplicate ratio greater ; and the resistance (by Law II and III) is as the motion communicated. Let us, therefore, see what motions arise from this law of resistance. SECTION II. 'If the motion of bodies that are resisted in tfie duplicate ratio of their velocities. PROPOSITION V. THEOREM III. Ff a body is resisted in the duplicate ratio of its velocity, and moves by its innate force only through a similar medium; and the times be taken in a geometrical progression, proceeding from less to greater terms : I say, that the velocities at the beginning of each of the times are in the same geometrical progression inversely ; and that the spaces are equal, which are described in each of the times. For since the resistance of the medium is proportional to the square of the velocity, and the decrement of the velocity is proportional to the resist ance : if the time be divided into innumerable equal particles, the squares of the velocities at the beginning of each of the times will be proportional to the differences of the same velocities. Let those particles of time be AK, KL, LM, &c., taken in the right line CD; and erect the perpendiculars AB, Kk, L/, Mm, &c., meeting the hyperbola BklmG, described with the centre C, and the rectangular asymptotes CD, CH. in B, kj I, m, (fee. ; then AB will be to Kk as CK to CA, and, by division, AB — Kk to Kk as AK C ARIMT to ^A> an(1 alternate^ AB — ^C to AK as Kk to CA ; and therefore as AB X Kk to AB X CA. Therefore since AK and AB X CA are given,* AB — Kk will be as AB X Kk ; and, lastly, when AB and KA* coincide, as AB2. And, by the like reasoning, KAr-U, J J-M/??, (fee., will be as Kk2. LI2, (fee. Therefore the squares of the lines AB, KA", L/, Mm, (fee., are as their differences ; and, therefore, since the squares of the velocities were shewn above to be as their differences, the progression of both will be alike. This being demonstrated it follows also that the areas described by these lines are in a like progres sion with the spaces described by these velocities. Therefore if the velo city at the beginning of the first time AK be expounded by the line AB, SEC. II.] OF NATURAL PHILOSOPHY. 25CJ and the velocity at the beginning of the second time KL by the line K& and the length described in the hrst time by the area AKA*B, all the fol lowing velocities will be expounded by the following lines \J. Mm, .fee. and the lengths described, by the areas K/, I mi. &c. And, by compo sition, if the whole time be expounded by AM, the sum of its parts, the whole length described will be expounded by AM/ftB the sum of its parts. Now conceive the time AM to be divided into the parts AK, KL, LM, (fee so that CA, CK, CL, CM, (fee. may be in a geometrical progression ; and those parts will be in the same progression, and the velocities AB, K/r, L/, Mm, (fee., will be in the same progression inversely, and the spaces de scribed Ak, K/, Lw, (fee., will be equal. Q,.E.D. COR. 1. Hence it appears, that if the time be expounded by any part AD of the asymptote, and the velocity in the beginning of the time by the ordinate AB, the velocity at the end of the time will be expounded by the ordinate DG ; and the whole space described by the adjacent hyperbolic area ABGD ; and the space which any body can describe in the same time AD, with the first velocity AB, in a non-resisting medium, by the rectan gle AB X AD. COR 2. Hence the space described in a resisting medium is given, by taking it to the space described with the uniform velocity AB in a non- resisting medium, as the hyperbolic area ABGD to the rectangle AB X AD. COR. 3. The resistance of the medium is also given, by making it equal, in the very beginning of the motion, to an uniform centripetal force, which could generate, in a body falling through a non-resisting medium, the ve locity AB in the time AC. For if BT be drawn touching the hyperbola in B. and meeting the asymptote in T, the right line AT will be equal to AC, and will express the time in which the first resistance, uniformly con tinned, may take away the whole velocity AB COR. 4. And thence is also given the proportion of this resistance to the force of gravity, or ay other given centripetal force. COR. 5. And, vice versa, if there is given the proportion of the resist- ; nee to any given centripetal force, the time AC is also given, in which c centripetal force equal to the resistance may generate any velocity as AB ; and thence is given the point B. through which the hyperbola, having CH CD for its asymptotes, is to be described : as also the space ABGD, which a body, by beginning its motion with that velocity AB, can describe in any time AD. in a similar resisting medium. PROPOSITION VI. THEOREM lVrc Homogeneous and equal spherical bodies, opposed hy resistances that are in the duplicate ratio of the velocities, and moving on by their innate force only, will, in times which are reciprocally as the velocities at thr. 260 THE MATHEMATICAL PRINCIPLES |BOOK II, v beg-in fiing, describe equal spaces, and lose parts of their velocities pro portional to the wholes. To the rectangular asymptotes CD, CH de scribe any hyperbola B6Ee, cutting the perpen diculars AB, rib, DE, de in B, b, E, e; let the initial velocities be expounded by the perpendicu lars AB, DE, and the times by the lines Aa, Drf. Therefore as Aa is to l)d, so (by the hypothesis) . is DE to AB, and so (from the nature of the hy- C "^ perbola) is CA to CD ; and, by composition, so is Crt to Cd. Therefore the areas ABba, DEerf, that is, the spaces described, are equal among themselves, and the first velocities AB, DE are propor tional to the last ab, de ; and therefore, by division, proportional to the parts of the velocities lost, AB — ab, DE — de. Q.E.D. PROPOSITION VII. THEOREM V. If spherical bodies are resisted in the duplicate ratio of their velocities, in times which are as the first motions directly, and the first resist ances inversely, they will lose parts of their motions proportional to the wholes, and will describe spaces proportional to those times and the first velocities conjunct It/. For the parts of the motions lost are as the resistances and times con junctly. Therefore, that those parts may be proportional to the wholes, the resistance and time conjunctly ought to be as the motion. Therefore the time will be as the motion directly and the resistance inversely. Where fore the particles of the times being taken in that ratio, the bodies will always loso parts of their motions proportional to the wholes, and there fore will retain velocities always proportional to their first velocities. And because of the given ratio of the velocities, they will always describe spaces which are as the first velocities and the times conjunctly. Q.E.D. COR. 1. Therefore if bodies equally swift are resisted in a duplicate ra tio of their diameters, homogeneous globes moving with any velocities whatsoever, by describing spaces proportional to their diameters, will lose parts of their motions proportional to the wholes. For the motion of each o-lobe will be as its velocity and mass conjunctly, that is, as the velocity and the cube of its diameter ; the resistance (by supposition) will be as the square of the diameter and the square of the velocity conjunctly ; and the •time (by this proposition) is in the former ratio directly, and in the latter inversely, that is, as the diameter directly and the velocity inversely ; and therefore' the space, which is proportional to the time and velocity is as the diameter. COR. 2. If bodies equally swift are resisted in a sesquiplicate ratio of their diameters, homogeneous globes, moving with any velocities whatso- SEC. 1L] OF NATURAL PHILOSOPHY. 261 ever, by describing spaces that are in a sesquiplicate ratio of the diameters, will lose parts of their motions proportional to the wholes. COR. 3. And universally, if equally swift bodies are resisted in the ratio of any power of the diameters, the spaces, in which homogeneous globes, moving with any velocity whatsoever, will lose parts of their motions pro portional to the wholes, will be as the cubes of the diameters applied to that power. Let those diameters be D and E : and if the resistances, where the velocities are supposed equal, are as T)n and E" ; the spaces in which the globes, moving with any velocities whatsoever, will lose parts of their motions proportional to the wholes, will be as D3 — n and E3 — n. And therefore homogeneous globes, in describing spaces proportional to D 3 — n and E3 — n, will retain their velocities in the same ratio to one another as at the beginning. COR. 4. Now if the globes are not homogeneous, the space described by the denser globe must be augmented in the ratio of the density. For the motion, with an equal velocity, is greater in the ratio of the density, and the time (by this Prop.) is augmented in the ratio of motion directly, and the space described in the ratio of the time. COR. 5. And if the globes move in different mediums, the space, in a medium which, cccteris paribus, resists the most, must be diminished in the ratio of the greater resistance. For the time (by this Prop.) will be di minished in the ratio of the augmented resistance, and the space in the ra tio of the time. LEMMA II. The moment of any genitum is equal to the moments of each of the gen- eratinrr sides drawn into the indices of the powers of those sides, and into their co-efficients continually. I call any quantity a genitum which is not made by addition or sub- duction of divers parts, but is generated or produced in arithmetic by the multiplication, division, or extraction of the root of any terms whatsoever : in geometry by the invention of contents and sides, or of the extremes and means of proportionals. Quantities of this kind are products, quotients, roots, rectangles, squares, cubes, square and cubic sides, and the like. These quantities I here consider as variable and indetermined, and increas ing or decreasing, as it were, by a perpetual motion or flux ; and I under stand their momentaneous increments or decrements by the name of mo ments ; so that the increments may be esteemed as added or affirmative moments ; and the decrements as subducted or negative ones. But take care not to look upon finite particles as such. Finite particles are not moments, but the very quantities generated by the moments. We are to conceive them as the just nascent principles of finite magnitudes. Nor do we in this Lemma regard the magnitude of the moments, but their firsf 262 THE MATHEMATICAL PRINCIPLES [BoOK 11 proportion, as nascent. It will be the same thing, if, instead of moments, we use either the velocities of the increments and decrements (which may also be called the motions, mutations, and fluxions of quantities), or any finite quantities proportional to those velocities. The co-efficient of any generating side is the quantity which arises by applying the genitum to ihat side. Wherefore the sense of the Lemma is, that if the moments of any quan tities A, B, C, &c., increasing or decreasing by a perpetual flux, or the velocities of the mutations which are proportional to them, be called a, 6, r, (fee., the moment or mutation of the generated rectangle AB will be «B -h b A ; the moment of the generated content ABC will be aBC -f b AC 4 -1 -2. .1 cAB; and the moments of the generated powers A2. A3, A4, A2, A2. A3, A*, A — ', A — 2, A— * will be 2aA, 3a A2, 4aA3, |«A — *, f«A* 11 3 i« A — s, |/iA — 3, — a A — 2, — 2aA — 3, — £aA — 2 respectively; and in general, that the moment of any power A^, will be ^ aAn-^. Also, that the moment of the generated quantity A2B will be 2aAB 4- bA~ ; the moment of the generated quantity A 3B4C2 will be 3«A2B4C2 + 4/>A3 A3 B3C2 4-2cA3B'C; and the moment of the generated quantity — or A »B — 2 will be 3aA 2 B — 2 — 2bA 3B — 3 ; and so on. The Lemma is thus demonstrated. CASE 1. Any rectangle, as AB, augmented by a perpetual flux, when, as yet, there wanted of the sides A and B half their moments \a and \b, was A — \a into B — \b, or AB — ±a B — \b A + \ab ; but as soon as the sides A and B are augmented by the other half moments, the rectangle be comes A 4- 4-a into B 4- \b, or AB -f ^a B 4- \b A -f \ab. From this rectangle subduct the former rectangle, and there will remain the exces.? aE -f bA. Therefore with the whole increments a and b of the sides, tin increment aB + f>A of the rectangle is generated. Q.K.D. CASE 2. Suppose AB always equal to G, and then the moment of the content ABC or GC (by Case 1) will be^C + cG, that is (putting AB and aB + bA for G and «*), aBC -h bAC 4- cAB. And the reasoning is the same for contents under ever so many sides. Q.E.D. CASE 3. Suppose the sides A, B, and C, to be always equal among them selves; and the moment «B + />A, of A2, that is, of the rectangle AB, will be 2aA ; and the moment aBC + bAC + cAB of A3, that is, of the content ABC, will be 3aA2. And by the same reasoning the moment of any power An is naAn — '. Q.E.D CASE 4. Therefore since -7 into A is 1, the moment of -r- drawn into A A SEC. 11.] OF NATURAL PHILOSOPHY. 263 A, together with — drawn into a. will be the moment of 1, that is, nothing. A Therefore the moment of -r, or of A — ', is -r— . And generally since A .A T- into An is I, the moment of —drawn into An together with — into A n A. A n naA" — ! will be nothing. And, therefore, the moment of -r- or A — n A will be— £T^~7- Q-E.D. V . t . ' i CASE 5. And since A2 into A2 is A, the moment of A1 drawn into 2 A2 will be a (by Case 3) ; and, therefore, the moment of A7 will be n~r~r or ^A-j £#A — £. And, generally, putting A~^ equal to B, then Am will be equal to Bn, and therefore maAm — ! equal to nbBn — ' , and ma A — ' equal to ?tbB — ', or tib A — ^ 5 an<i therefore ri a A —^~ is equal to &, that is, equal to the moment of A^. Q.E.D. CASE 6. Therefore the moment of any generated quantity AmBn is the moment of Am drawn into Bn, together with the moment of Bn drawn into A", that is, maAm — ' B" -f- nbBn — ! Am; and that whether the indices in arid n of the powers be whole numbers or fractions, affirmative or neg ative. And the reasoning is the same for contents under more powers. Q.E.D. COR. 1. Henoe in quantities continually proportional, if one term is given, the moments of the rest of the terms will be as the same terms mul tiplied by the number of intervals between them and the given term. Let A, B, C, D; E, F, be continually proportional ; then if the term C is given, the moments of the rest of the terms will be among themselves as — 2A, — B? D, 2E, 3F. COR. 2. And if in four proportionals the two means are given, the mo ments of the extremes will be as those extremes. The same is to be un derstood of the sides of any given rectangle. COR. 3. And if the sum or difference of two squares is given, the mo ments of the sides will be reciprocally as the sides. SCHOLIUM. In a letter of mine to Mr. /. Collins, dated December 10, 1672, having described a method of tangents, which I suspected to be the same with Slusius*s method, which at that time wag not made public, I subjoined these words • This is one particular, or rather a Corollary, of a general nte 264 THE MATHEMATICAL PRINCIPLES [BjOK II. thod, which extends itself, without any troublesome calculation, not ojdy to the drawing of tangents to any curve lines, whether geometrical or mechanical, or any how respecting right lines or other cnrves, but also to the resolving other abstrnser kinds of problems about the crookedness, areas, lengths, centres of gravity of curves, &c. ; nor is it (as Hudd^ri's method de Maximis & Minimia) limited to equations which are free from surd quantities. This method I have interwoven with that other oj working in equations, by reducing them to infinite serie?. So far that letter. And these last words relate to a treatise I composed on that sub ject in the year 1671. The foundation of that general method is contain ed in the preceding Lemma. PROPOSITION VIII. THEOREM VI. If a body in an uniform medium, being uniformly acted upon by the force of gravity, ascends or descends in a right line ; and the whole space described be distinguished into equal parts, and in the beginning of each of the parts (by adding or subducting the resisting force of the medium to or from the force of gravity, when the body ascends or de scends] yon collect the absolute forces ; I say, that those absolute forces ire in a geometrical progression. For let the force of gravity be expounded by the given line AC ; the force of resistance by the indefi nite line AK ; the absolute force in the descent of the body by the difference KC : the velocity of the I tody <^LKJL&i>F/ by a line AP, which shall be a mean proportional be tween AK and AC, and therefore in a subduplicate ratio of the resistance; the increment of the resistance made in a given particle of time by the li- neola KL, and the contemporaneous increment of the velocity by the li- neola PQ ; and with the centre C, and rectangular asymptotes CA, CH, describe any hyperbola BNS meeting the erected perpendiculars AB, KN, LO in B, N and O. Because AK is as AP2 , the moment KL of the one will be as the moment 2APQ of the other, that is, as AP X KC ; for the in crement PQ of the velocity is (by Law II) proportional to the generating force KC. Let the ratio of KL be compounded with the ratio KN, and the rectangle KL X KN will become as AP X KC X KN ; that is (because the rectangle KC X KN is given), as AP. But the ultimate ratio of the hyperbolic area KNOL to the rectangle KL X KN becomes, when the points K and L coincide, the ratio of equality. Therefore that hyperbolic evanescent area is as AP. Therefore the whole hyperbolic area ABOL is composed of particles KNOL which are always proportional to the velocity AP; and therefore is itself proportional to the space described with that velocity. Let ,that area be now divided into equal parts SEC. IJ.J OF NATURAL PHILOSOPHY. 265 as ABMI, IMNK, KNOL, (fee., and the absolute forces AC, 1C, KC, LC, (fee., will be in a geometrical progression. Q,.E.D. And by a like rea soning, in the ascent of the body, taking, on the contrary side of the point A, the equal areas AB?m, i/nnk, knol, (fee., it will appear that the absolute forces AC. iG, kC, 1C, (fee., are continually proportional. Therefore if all the spaces in the ascent and descent are taken equal, all the absolute forces 1C, kC, iC, AC, 1C, KC, LC, (fee., will be continually proportional. Q,.E.D. COR. 1. Hence if the space described be expounded by the hyperbolic area ABNK, the force of gravity, the velocity of the body, and the resist ance of the medium, may be expounded by the lines AC, AP, and AK re spectively • and vice versa. COR. 2. And the greatest velocity which the body can ever acquire in an infinite descent will be expounded by the line AC. COR. 3. Therefore if the resistance of the medium answering to any given velocity be known, the greatest velocity will be found, by taking it to that given velocity in a ratio subduplicate of the ratio which the force of gravity bears to that known resistance of the medium. PROPOSITION IX. THEOREM VII. Supposing ivhat is above demonstrated, I say, that if the tangents of t-he angles of the sector of a circle, and of an hyperbola, be taken propor tional to the velocities, the radius being of a fit magnitude, all the time of the ascent to the highest place icill be as the sector of the circle, and all the time of descending from the highest place as the sector of t/ie hyperbola. To the right line AC, which ex presses the force of gravity, let AD drawn perpendicular and equal. From the centre D with the semi-diameter AD describe as well the cmadrant A^E -t of a circle, as the rectangular hyper bola AVZ, whose axis is AK, principal vertex A, and asymptote DC. Let Dp, DP be drawn ; and the circular sector AtD will be as all the time of the as cent to the highest place ; and the hy perbolic sector ATD as all the time of descent from the highest place; ii BO be that the tangents Ap, AP of those sectors be as the velocities. CASE 1. Draw Dvq cutting off the moments or least particles tDv and ^ ?, described in the same time, of the sector ADt and of the triangle AD/?. Since those particles (because of the common angle D) are in a du plicate ratio of the sides, the particle tDv will be as — -^-^-— , that is 266 THE MATHEMATICAL PRINCIPLES [BOOK li. (because tD is given), as ^f. But joD8 is AD3 + Ap2, that is, AD2 -h AD X AA-, or AD X Gk ; and (/Dp is 1 AD X pq. Therefore tDv, the BO particle of the sector, is as ^ , ; that is, as the least decrement pq of the velocity directly, and the force Gk which diminishes the velocity, inversely ; and therefore as the particle of time answering to the decrement of the ve locity. And, by composition, the sum of all the particles tDv in the sector AD/ will be as the sum of the particles of time answering to each of the lost particles pq of the decreasing velocity Ap, till that velocity, being di minished into nothing, vanishes; that is, the whole sector AD/ is as the whole time of ascent to the highest place. Q.E.D. CASE 2. Draw DQV cutting off the least particles TDV and PDQ of the sector DAV, and of the triangle DAQ ; and these particles will be to each other as DT2 to DP2, that is (if TX and AP are parallel), as DX2 to DA2 or TX2 to AP2 ; and, by division, as DX2 — TX2 to DA2 - AP2. But. from the nature of the hyperbola, DX2 — TX2 is AD2 ; and, by the supposition, AP2 is AD X AK. Therefore the particles are to each other as AD2 to AD2 — AD X AK ; that is, as AD to AD — AK or AC ; and to CK : and therefore the particle TDV of the sector is - PQ therefore (because AC and AD are given) as CK that is, as the increment of the velocity directly, and as the force generating the increment inverse ly ; and therefore as the particle of the time answering to the increment. And, by composition, the sum of the particles of time, in which all the par ticles PQ of the velocity A I3 are generated, will be as the sum of the par ticles of the sector ATI) ; that is, the whole time will be as the whole sector. Q.E.D. COR. 1. Hence if AB be equal to a fourth part of AC, the space which a body will describe by falling in any time will be to the space which the body could de scribe, by moving uniform]}' on in the same time with its greatest velocity AC, as the area ABNK, which ex presses the space described in falling to the area ATD, which expresses the time. For since AC is to AP as AP _ to AK, then (by Cor. 1, Lem. II, of this Book) LK is to PQ as 2AK to AP, that is, as 2AP to AC, and thence LK is to ^PQ as AP to JAC or AB ; and KN is to AC or AD as AB tc . II.] OF NATURAL PHILOSOPHY. 267 UK ; and therefore, ex cequo, LKNO to DPQ, as AP to CK. But DPQ was to DTV as CK to AC. Therefore, ex aquo, LKNO is to DTV r,? AP to AC ; that is, as the velocity of the falling body to the greatest velocity which the body by falling can acquire. Since, therefore, the moments LKNO and DTV of the areas ABNK and ATD are as the ve locities, all the parts of those areas generated in the same time will be as the spaces described in the same time ; and therefore the whole areas ABNK and ADT, generated from the beginning, will be as the whole spaces de scribed from the beginning of the descent. Q.E.D. COR. 2. The same is true also of the space described in the ascent. That is to say, that all that space is to the space described in the same time, with the uniform velocity AC, as the area ABttk is to the sector ADt. COR. 3. The velocity of the body, falling in the time ATD, is to the velocity which it would acquire in the same time in a non-resisting space, as the triangle APD to the hyperbolic sector ATD. For the velocity in a non-resisting medium Avould be as the time ATD, and in a resisting me dium is as AP, that is, as the triangle APD. And those velocities, at the beginning of the descent, are equal among themselves, as well as those areas ATD, APD. COR. 4. By the same argument, the velocity in the ascent is to the ve locity with which the body in the same time, in a non-resisting space, would lose all its motion of ascent, as the triangle ApD to the circular sector AtD ; or as the right line Ap to the arc At. COR. 5. Therefore the time in which a body, by falling in a resisting medium, would acquire the velocity AP, is to the time in which it would acquire its greatest velocity AC, by falling in a non-resisting space, as the sector ADT to the triangle ADC : and the time in which it would lose its velocity Ap, by ascending in a resisting medium, is to the time in which it would lose the same velocity by ascending in a non-resisting space, as the arc At to its tangent Ap. COR. 6. Hence from the given time there is given the space described in the ascent or descent. For the greatest velocity of a body descending in wfinitum is given (by Corol. 2 and 3, Theor. VI, of this Book) ; and thence the time is given in which a body would acquire that velocity by falling in a non-resisting space. And taking the sector ADT or ADt to the tri angle ADC in the ratio of the given time to the time just now found, there will be given both the velocity AP or Ap, and the area ABNK or AB//&, which is to the sector ADT, or AD/, as the space sought to the space which would, in the given time, be uniformly described with that greatest velocity found just before. COR. 7. And by going backward, from the given space of ascent or de scent AB»?A: or ABNK, there will be given the time AD* or ADT. 268 THE MATHEMATICAL PRINCIPLES [BOOK ii PROPOSITION X. PROBLEM III. Suppose the uniform force of gravity to tend directly to the plane of the horizon, and the resistance to be as the density of the medium and the square of the velocity coiijunctly : it is proposed to find the density of the medium in each place, which shall make the body move in any given carve line ; the velocity of the body and the resistance of the medium in each place. Let PQ be a plane perpendicular to the plane of the scheme itself; PFHQ a curve line meeting that plane in the points P and Q ; G, H, I, K four places of the body going on in this \ curve from F to Q ; and GB, HC, ID, KE four parallel ordinates let fall p A. 33 c^D E Q from these points to the horizon, and standing on the horizontal line PQ at the points B, C, D, E ; and let the distances BC, CD, DE, of the ordinates be equal among themselves. From the points G and H let the right lines GL, HN, be drawn touching the curve in G and H, and meeting the ordinates CH, DI, produced upwards, in L and N : and complete the parallelogram HCDM. And the times in which the body describes the- arcs GH, HI, will be in a subduplicate ratio of the altitudes LH, NI; which the bodies would describe in those times, by falling from the tangents; and the velocities will be as the lengths de scribed GH, HI directly, and the times inversely. Let the times be ex- C*TT TTT pounded by T and t, and the velocities by —•=- and ---; and the decrement J_ L /-^TT TTT of the velocity produced in the time t will be expounded by -7^ — . This decrement arises from the resistance which retards the body, and from the gravity which accelerates it. Gravity, in a falling body, which in its fall describes the space NI, produces a velocity with which it would be able to describe twice that space in the same time, as Galileo has demonstrated ; 2NI that is, the velocity : but if the body describes the arc HI, it augments MIxNl HI — ; and therefore generates that arc only by the length HI — HN or only the velocity — iff—- Let this velocity be added to the before- t X H.JL mentioned decrement, and we shall have the decrement of the velocity GH HI SMI X Nl arising from the resistance alone, that is, -^ : — h T SEC. II.J OF NATURAL PHILOSOPHY. 269 2NI. Therefore since, in the same time, the action of gravity generates, in a fall ing body, the velocity , the resistance will be to the gravity as 7^ HI 2MI X NI 2NI t X GH 2MI X NI — + — TTT- to — or as — ^ — — HI -f Now for the abscissas CB, CD, CE, put — o, o, 2o. For the ordinate CH put P ; and for MI put any series Qo + Ro2 + So3 +, &c. And all the terms of the series after the first, that is, Ro2 + So3 +, (fee., will be NI ; and the ordinates DI, EK, and BGwill be P — Qo—Ro2— So3— p A B c T> E (fee., P — 2Qo— 4Ro2— SSo3— , (fee., and P -\- Qo — Ro2 + So3—, (fee., respectively. And by squaring the differences of the ordinates BG — CH and CH — DI, and to the squares thence produced adding the squares of BC and CD themselves, you will have oo -f- QQoo — 2QRo3 +, (fee., and oo -f QQoo -f 2QRo3 +, (fee., the squares of the arcs GH, HI ; whose QRoo QRoo roots o y - • , and o </! 4- QQ 4- — are the 1 + QQ v/1 + QQ s/1 -f QQ arcs GH and HI. Moreover, if from the ordinate CH there be subducted half the sum of the ordinates BG and DI, and from the ordinate DI there be subducted half the sum of the ordinates CH and EK, there will remain Roo and Roo + 3So3, the versed sines of the arcs GI and HK. And these are proportional to the lineolae LH and NI, and therefore in the duplicate ratio of the infinitely small times T and t : and thence the ratio ~, is ^ R + SSo R -f ^ or So , t X GH TTT 2MI X NI , — : and — ^ HI H TTT , by substituting R T HI the values of ™, GH, HI, MI and NI just found, becomes -^- J- /w-Lt/ I + QQ. Arid since 2NI is 2Roo, the resistance will be now to the OO gravity as -- TT'Q to 2Roo> that is> as 3S r to 4RR. And the velocity will be such, that a body going off therewith from any place H, in the direction of the tangent HN, would describe, in vacuo, a parabola, whose diameter is HC, and its latus rectum NT or -- ^ ---- . And the resistance is as the density of the medium and the square of the velocity conjunctly ; and therefore the density of the medium is as the resistance directly, and the square of the velocity inversely ; that is, as 270 THE MATHEMATICAL PRINCIPLES [BOOK II. QQ __ 4RR Q.E.I. COR. 1. If the tangent HN be produced both ways, so as to meet any HT ordinatc AF in T - will be equal to V/T+ QQ; and therefore in what has gone before may be put for ^ \ -\- QQ. By this means the resistance will be to the gravity as 3S X HT to 4RR X AC ; the velocity will be a* •r-pj — --^, and the density of the medium will be as „— -TT-n. AO -v/ ±i Jti X H 1 COR. 2. And hence, if the curve line PFHQ be denned by the relation between the base or abscissa AC and the ordinate CH, as is usual, and the value of the ordinate be resolved into a converging series, the Problem will be expeditiously solved by the first terms of the series ; as in the fol lowing examples. EXAMPLE 1. Let the line PFHQ, be a semi-circle described upon the diameter PQ, to find the density of the medium that shall make a projec tile move in that line. Bisect the diameter PQ in A ; and call AQ, n ; AC, a ; CH, e ; and CD, o ; then DI2 or AQ,2 — AD 2 = nn — aa — 2ao — oo, or ec. — 2ao — oo ; and the root being extracted by our method, will give DI = e — ao oo aaoo ao3 a3 o3 ~e~~~2e 2e? ~~~ W ~2? ' &C* Here put nn f°r ee + aa> and ao nnoo anno3 DI will become = e — , &c. e 2e3 2e5 Such series I distinguish into successive terms after this manner : I call that the first term in which the infinitely small quantity o is not found ; the second, in which that quantity is of one dimension only ; the third, in which it arises to two dimensions ; the fourth, in which it is of three ; and so ad infinitum. And the first term, which here is e, will always denote the length of the ordinate CH, standing at the beginning of the indefinite quantity o. The second term, which here is — , will denote the difference between CH and DN ; that is, the lineola MN which is cut off by com pleting the parallelogram HC DM; and therefore always determines the CM? position of the tangent HN ; as, in this case, by taking MN to HM as — G to o, or a to e. The third term, which here is -£—, will represent the li neola IN, which lies between the tangent and the curve ; and therefore determines the angle of contact IHN, or the curvature which the curve line SEC. II.] OF NATURAL PHILOSOPHY. 271 has in H. If that lineola IN is of a finite magnitude, it will be expressed by the third term, together with those that follow in wfinitu:.:i. But if that lineola be diminished in. infini- tnm, the terms following become in finitely less than the third term, and therefore may be neglected. The fourth term determines the variation of the curvature ; the fifth, the varia tion of the variation ; and so on. Whence, by the way, appears no con-p~" ~K B~C~D~E~ temptible use of these series in the solution of problems that depend upon tangents, and the curvature of curves. ao 77/700 anno 3 Now compare the series e — ^— ^~ &c., with the e Ze3 Ze* series P — Qo — - Roo — So3 — &c., and for P, Q, II and S? put e, -, ^-^ and ~ — , and for ^ 1 + QQ put 1 H or - ; and the density oi 2e 5 ' ee e ' the medium will come out as — ; that is (because n is given), as - or lie ' e ~Yj, that is, as that length of the tangent HT, which is terminated at the OH. semi-diameter AF standing perpendicularly on PQ : and the resistance will be to the gravity as 3a to 2>/, that is, as SAC to the diameter PQ of the circle; and the velocity will be as i/ CH. Therefore if the body goes from the place F, with a due velocity, in the direction of a line parallel to PQ, and the density of the medium in each of the places H is as the length of the tangent HT, and the resistance also in any place H is to the force of gravity as SAC to PQ, that body will describe the quadrant FHQ of a circle. Q.E.I. But if the same body should go-frorn the place P, in the direction of a line perpendicular to PQ, and should begin to move in an arc of the semi circle PFQ, we must take AC or a on the contrary side of the centre A ; and therefore its sign must be changed, and we must put — a for + a. Then the density of the medium would come out as . But nature does not admit of a negative density, that is, a density which accelerates the motion of bodies; and therefore it cannot naturally come to pass that a body by ascending from P should describe the quadrant PF of a circle. To produce such an effect, a body ought to be accelerated by an impelling medium, and not impeded by a resisting one. EXAMPLE 2. Let the line PFQ be a parabola, having its axis AF per- 272 THE MATHEMATICAL PRINCIPLES [BOOK IL pendicular to the horizon PQ, to find the density of the medium, which will make a projectile move in that line. From the nature of the parabola, the rectangle PDQ, is equal to the rectangle under the ordinate DI and some given right line ; that is, if that right line be called b ; PC, a; PQ, c; CH, e; and CD, o; the rectangle a A. CD ~Q + o into c — a — o or ac — aa — 2ao -{-co — oo, ia ac — aa equal to the rectangle b into DI, and therefore DI is equal to -- 7 -- h c — 2a oo c — 2a — -. — o — —r. Now the second term — -, — o of this series is to be put b b b oo for Q,o, and the third term -r for Roo. But since there are no more terms, the co-efficient S of the fourth term will vanish ; and therefore the S ouantitv - — , to which the density of the medium is proper- ' R v i tional, will be nothing. Therefore, where the medium is of no density, the projectile will move in a parabola ; as Galileo hath heretofore demon strated. Q.E.I. EXAMPLE 3. Let the line AGK be an hyperbola, having its asymptote NX perpendicular to the horizontal plane AK, to find the density of the medium that will make a projectile move in that line. Let MX be the other asymptote, meeting the ordinate DG produced in V ; and from the nature of the hyperbola, the rectangle of XV into VG will be given. There is also given the ratio of DN to VX, and therefore the rectangle of DN into VG is given. Let that be bb : and, completing the parallelo gram DNXZ, let BN be called a; BD, o; NX, c; and let the given ratio of VZ to ZX or DN be -. Then DN will be equal MA. BD K N bb to a — o} VG equal to , VZ equal to — X a — o. and GD or NX a — o' n m m -VZ— VG equal to c— — a + — o — . Let the term - — be n n a — o a — o bb bb bb bb , resolved into the converging series ~^" + ^° + ^l00 + ^4 °> &c'» and m bb m bb bb bb GD will become equal to c — - a— - + -o — ~ o — ^ o2 — 51 ° SEC. II.] OF NATURAL PHILOSOPHY. 273 &c. The second term — o o of this series is to be u?ed for Qo; the n aa third — o2, with its sign changed for Ro2 ; and the fourth — o3, with its m bb bb bb sign changed also for So3, and their coefficients , — and — are to n aa a a be pat for Q, R, and S in the former rule. Which being done, the den- bb a* sity of the medium will come out as , , bb a mm nn 2mbb naa I mm -— , that is, if in VZ you take VY equal to aa aa 1 m2 VG, as YT7- For aa and —^ a 2 2mbb b' nn n aa 2mbb b 4 — • H are the squares of XZ n aa and ZY. But the ratio of the resistance to gravity is found to be that of 3XY to 2YG ; and the velocity is that with which the body would de- XY2 scribe a parabola, whose vertex is G, diameter DG, latus rectum ^v . Sup pose, therefore, that the densities of the medium in each of the places G are reciprocally as the distances XY, and that the resistance in any place G is to the gravity as 3XY to 2YG ; and a body let go from the place A, with a due velocity, will describe that hyperbola AGK. Q.E.I. EXAMPLE 4. Suppose, mdeMtely, the line AGK to be an hyperbola described with the centre X, and the asymptotes MX, NX, so that, having constructed the rectangle XZDN, whose side ZD cuts the hyperbola in G and its asymptote in V, VG may be reciprocally as any power DNn of the line ZX or DN, whose index is the number n : to find the density of the medium in which a projected body will describe this curve. For BN, BD, NX, put A, O, C, respec- ^ tively, and let VZ be to XZ or DN as d to bb e, and VG be equal to be equal to A — O, VG == ^= then DN will VZ = O, and GD or NX — VZ — VG equal 274 term THE MATHEMATICAL PRINCIPLES [BOOK H nbb nn -f- n bb U ! J • x • £ • *° •=rr be resolved into an infinite series -r- + — A — Of A" A.n 3 -±- 3nn + 2/i X O + n " ~ x bb O3,&c,,andGD will be equal g^TT-T X 00 O2 + c£ bb d nbb + ?m - toC — -A--T-+-O- -r O - ~ e A" e A" + l 2An -f + H i^T 't"\ bb°3> &c- The second tcrm - O - -T 6An + e An 4- l series is to be used for 0,0, the third ^ 0 66O2 for Roo, the fourth — -£\~r~3 bbO5 for So3. And thence the density of the medium Oof this -, in any place G7 will be 2dnbb nub*' and therefore if in VZ you take VY equal to n X VG, that density is re- n w IT- j ^ * 2rf//66 /mfi4 ciprocally as XY. For A2 and — A2 -- — A + — r are the tc/ o^x ./\_ " squares of XZ and ZY. Hut the resistance in the same place G is to the force of gravity as 3S X - to 4RR, that is, as XY to And the velocity there is the same wherewith the projected body would move in a parabola, whose vertex is G, diameter GD, and latus rectum 2XY2 or - -------- - - . Q.E.I. R nn VG AC HT SCHOLIUM. In the same manner that the den sity of the medium comes out to be as S X AC . « Tjr™ m ^°r- 1) if the resistance lx X HI is put as any power V" of the velocity V, the density of the medium will come out to be as B C D E Q — . x S And therefore if a curve can be found, such that the ratio of to 4 — o R— i- SEC. II.J OF NATURAL PHILOSOPHY, 275 n — 1 , or ofgr^ to may be given ; the body, in an uni- z HT AC form medium, whose resistance is as the power V" of the velocity V, will move in this curve. But let us return to more simple curves. Because there can be no motion in a para bola except in a non-resisting medium, but in the hyperbolas here described it is produced by a perpetual resistance ; it is evident that the line which a projectile describes in an uniformly resisting medium approaches nearer to these hyperbolas than to a parabola. That line is certainly of the hyperbolic kind, but about the vertex it is more distant from the asymptotes, and in the parts remote from the vertex draws nearer to them than these hy- M JL BD K perbolas here described. The difference, however, is not so great between the one and the other but that these latter may be commod^ously enough used in practice instead of the former. And perhaps these may prove more useful than an hyperbola that is more accurate, and at the same time more compounded. They may be made use of, then, in this manner. Complete the parallelogram XYGT, and the right line GT will touch the hyperbola in G, and therefore the density of the medium in G is re- GT2 ciprocally as the tangent GT, and the velocity there as ^ -^=- ; and the resistance is to the force of gravity as GT to Therefore if a body projected from the place A, in the direction of the right line AH, describes the hyperbola AGK and AH produced meets the asymptote NX in H, arid AI drawrri parallel to it meets the other asymptote MX in I ; the density of the mediu.n in A will be reciprocally as AH. and the velocity of the body as -J AH1 . . . and the resis'an^e there to the force Al 2nn n +2 ^- X GV. . TT 2nn + 2n of gravity rs AH to ZiTo ~ X AI. Her,ce the following rules a e deduced. RULE 1. If the density of the medium at A, and the velocity with which the body is projected remain the same, and the angle NAH be changed , the lengths AH, AI, HX will remain. Therefore if those lengths, in any 276 THE MATHEMATICAL PRINCIPLES [BOOK II. one case, are found, the hyperbola may afterwards be easily determined from any given angle NAH. RULE 2. If the angle NAH, and the density of the medium at A, re main the same, and the velocity with which the body is projected be changed, the length AH will continue the same ; and AI will be changed in a duplicate ratio of the velocity reciprocally. RULE 3. If the angle NAH, the velocity of the body at A, and the ac- celerative gravity remain the same, and the proportion of the resistance at A to the motive gravity be augmented in any ratio ; the proportion of AH to A I will be augmented in the same ratio, the latus rectum of the above- AH2 mentioned parabola remaining the same, and also the length propor- AI tional to it ; and therefore AH will be diminished in the same ratio, and AI will be diminished in the duplicate of that ratio. But the proportion of the resistance to the weight is augmented, when either the specific grav ity is made less, the magnitude remaining equal, or when the density of the medium is made greater, or when, by diminishing the magnitude, the resistance becomes diminished in a less ratio than the weight. RULE 4. Because the density of the medium is greater near the vertex of the hyperbola than it is in the place A, that a mean density may be preserved, the ratio of the least of the tangents GT to the tangent AH ought to be found, and the density in A augmented in a ratio a little greater than that of half the sum of those tangents to the least of the tangents GT. RULE 5. If the lengths AH, AI are given, and the figure AGK is to be described, produce HN to X, so that HX may be to AI as n -\- 1 to 1 ; and with the centre X, and the asymptotes MX, NX, describe an hyperbola through the point A, such that AI may be to any of the lines VG as XV" to xr. RULE 6. By how much the greater the number n is, so much the more accurate are these hyperbolas in the ascent of the body from A, and less accurate in its descent to K ; and the contrary. The conic hyperbola keeps a mean ratio between these, and is more simple than the rest. There fore if the hyperbola be of this kind, and you are to find the point K, where the projected body falls upon any right line AN passing through the point A, let AN produced meet the asymptotes MX, NX in M and N, and take NK equal to AM. RULE 7. And hence appears an expeditious method of determining this hyperbola from the phenomena. Let two similar and equal bodies be pro jected with the same velocity, in different angles HAK, h A k, and let them fall upon the plane of the horizon in K and k ; and note the proportion of AK to A A". Let it be as d to e. Then erecting a perpendicular A I of uny length, assume any how the length AH or Ah, and thence graphically, SEC. II. OF NATURAL PHILOSOPHY. 27? or by scale and compass, collect the lengths AK, A/>* (by Rule 6). If the ratio of AK to A/.* bo the same with that of d to e, the length of AH was rightly assumed. If not, take on the indefinite right line SM, the length SM equal to the assumed AH ; and erect a perpendicular MN equal to the AK d difference -r-r of the ratios drawn into any given right line. By the like method, from several assumed lengths AH, you may find several points N ; and draw througli them all a regular curve NNXN, cutting tr.e right line SMMM in X. Lastly, assume AH equal to the abscissa SX, and thence find again the length AK ; and the lengths, w'hich are to the as sumed length AI, and this last AH, as the length AK known by experi ment, to the length AK last found, will be the true lengths AI and AH, which were to be found. But these being given, there will be given also the resisting force of the medium in the place A, it being to the force of gravity as AH to JAI. Let the density of the medium be increased by Rule 4, and if the resisting force just found be increased in the same ratio, it will become still more accurate. RULE 8. The lengths AH, HX being found ; let there be now re quired the position of the line AH, according to which a projectile thrown with that given velocity shall fall upon any point K. At the joints A and K, erect the lines AC, KF perpendicular to the horizon : whereof let AC be drawn downwards, and be equal to AI or ^HX. With the asymp totes AK, KF, describe an hyperbola, whose conjugate shall pass through the point C ; and from the centre A, with the interval AH. describe a cir cle cutting that hyperbola in the point H ; then the projectile thrown in the direction of the right line AH will fall upon the point K. Q.E.I. For the point H, because of the given length AH, must be somewhere in the circumference of the described circle. Draw CH meeting AK and KF in E and F: and because CH, MX are parallel, and AC, AI equal, AE will be equal to AM, and therefore also equal to KN. But CE is to AE as FH to KN. and therefore CE and FH are equal. Therefore the point H falls upon the hyperbolic curve described with the asymptotes AK,.KF whose conjugate passes through the point C ; and is therefore found in the 27S THE MATHEMATICAL PRINCIPLES [BOOK 11 common intersection of this hyperbolic curve and the circumference of the de scribed circle. Q.E.D. It is to be ob served that this operation is the same, whether the right line AKN be parallel to the horizon, or inclined thereto in any an gle : and that from two intersections H, //., there arise two angles NAH, NAA ; and that in mechanical practice it is suf ficient once to describe a circle, then to apply a ruler CH, of an indeterminate length, HO to the point C, that its part PH, intercepted between the circle and the right line FK, may bo equal to its part CE placed between the point C and the right line AK What has been said of hyperbolas may be easily applied to pir i'>;>l.i3. For if a parabola be re presented by XAGK, touched by a right line XV in the vertex X, and the ordinates IA, YG be as any powers XI", XV"; of the abscissas XI, XV ; draw XT, GT, AH, whereof let XT be parallel to VG, and let GT, AH touch the parabola in G and A : and a body projected from any place A, in the direction of the right line AH, with a due velocity, will describe this parabola, if the density of the medium in each of the places G be reciprocally as the tangent GT. In that case the velocity in G will be the same as would cause a body, moving in a non- resisting space, to describe a conic parabola, having G for its vertex, VG 2GT2 produced downwards for its diameter, and -. — for its latus nn — n X VG rectum. And the resisting force in G will be to the force of gravity as GT to 2nti — 2tt ~2~ VG. Therefore if NAK represent an horizontal line, and botli the density of the medium at A, and the velocity with which the body is projected, remaining the same, the angle NAH be any how altered, the lengths AH, AI, HX will remain; and thence will be given the vertex X of the parabola, and the position of the right line XI ; and by taking VG to IA as XVn to XI", there will be given all the points G of the parabola, through which the projectile will pass. SEC. IILJ OF NATURAL PHILOSOPHY. 279 SECTION III. Of the motions of bodies which are resisted partly in the ratio of the ve locities, and partly in the duplicate of the same ratio. PROPOSITION XI. THEOREM VIII. If a body be resisted partly in the ratio and partly in the duplicate ratio of its velocity, and moves in a similar medium by its innate force only; and the times be taken in arithmetical progression; then quantities reciprocally proportional to the velocities, increased by a cer tain given quantity, will be in geometrical progression. With the centre C, and the rectangular asymptotes ^ OADd and CH, describe an hyperbola BEe, and let | \p AB, DE, de. be parallel to the asymptote CH. In | the asymptote CD let A, G be given points ; and if the time be expounded by the hyperbolic area ABED uniformly increasing, I say, that the velocity may ~r be expressed by the length DF, whose reciprocal GD, together with the given line CG, compose the length CD increasing in a geometrical progression. For let the areola DEec/ be the least given increment of the time, and Dd will be reciprocally as DE, and therefore directly as CD. Therefore the decrement of ^TR, which (by Lem. II, Book II) is ^no, will be also as D tf CD CG + GD 1 CG GO* °r — GD2 ~ ' fc 1S>aS GD + GJD2* *nerefore tne timc uniformly increasing by the addition of the given particles EDcfe, it fol lows that r decreases in the same ratio with the velocity. For the de crement of the velocity is as the resistance, that is (by the supposition), as the sum of two quantities, whereof one is as the velocity, and the other as the square of the velocity ; and the decrement of ~~ is as the sum of the 1 C^(^ 1 quantities ~-^=r and pfp,> whereof the first is ^^r itself, and the last i i is a* /-TFT; • therefore T^-R is as tne velocity, the decrements of both -' CilJ being analogous. And if the quantity GD reciprocally proportional to T, be augmented by the given quantity CG ; the sum CD, the time ABED uniformly increasing, will increase !n a geometrical progression. Q.E.D. THE MATHEMATICAL PRINCIPLES [BOOK II COR. 1. Therefore, if, having the points A and G given, the time bo expounded by the hyperbolic area ABED, the velocity may be expounded by -r the reciprocal of GD. COR. 2. And by taking GA to GD as the reciprocal of the velocity at the beginning to the reciprocal of the velocity at the end of any time ABED, the point G will be found. And that point being found the ve locity may be found from any other time given. PROPOSITION XII. THEOREM IX. The same things being supposed, I say, that if the spaces described are taken in arithmetical progression, the velocities augmented by a cer tain given quantity will be in geometrical progression. In the asymptote CD let there be given the point R, and, erecting the perpendicular RS meeting the hyperbola in S, let the space de scribed be expounded by the hyperbolic area I RSED ; and the velocity will be as the length J GD, which, together with the given line CG, ** composes a length CD decreasing in a geo metrical progression, while the space RSED increases in an arithmetical [(regression. For, because the incre nent EDde of the space is given, the lineola DC?, which is the decrement of GD, will be reciprocally as ED, and therefore directly as CD ; that is, as the sum of the same GD and the given length CG. But the decrement of the velocity, in a time reciprocally propor tional thereto, in which the given particle of space D^/eE is described, is as the resistance and the time conjunctly, that is. directly as the sum of two quantities, whereof one is as the velocity, the other as the square of the velocity, and inversely as the veh city ; and therefore directly as the sum of two quantities, one of which is given, the other is- as the velocity. Therefore the decrement both of the velocity and the line GD is as a given quantity and a decreasing quantity conjunctly; and, because the decre ments are analogous, the decreasing quantities will always be analogous ; viz., the velocity, and the line GD. U.E.D. COR. 1. If the velocity be expounded by the length GD, the space de scribed will be as the hyperbolic area DESR. COR. 2. And if the point „ be assumed any how, the point G will be found, by taking GR to GD as the velocity at the beginning to the velo city after any space RSED is described. The point G being given, the space is given from the given velocity : and the contrary. Cotw 3. Whence since (by Prop. XI) the velocity is given from the given SEC. Ilt.1 Or NATURAL PHILOSOPHY. 281 time, and (by this Prop.) the space is given from the given velocity ; the space will be given from the given time : and the contrary. PROPOSITION XKI. THEOREM X. Supposing that a body attracted downwards by an uniform gravity as cends or descends in a right line; and that the same is resisted partly in the ratio of its velocity, and partly in the duplicate ratio thereof: I say, that, if right lines parallel to the diameters of a circle and an hyperbola, be drawn through the ends of the, conjugate diame ters, and the velocities be as some segments of those parallels drawn from a given point, the times will be as the sectors of the, areas cut off by right lines drawn from the centre to the ends of the segments ; and the contrary. CASE 1. Suppose first that the body is ascending, and from the centre D, with any semi-diameter DB, describe a quadrant BETF of a circle, and through the end B of the semi-diameter DB draw the indefi nite line BAP, parallel to the semi-diameter DF. In chat line let there be given the point A, and take the segment AP proportional to the velocity. And since one part of the resistance is as the velocity, and another part as the square of the velocity, let the whole resistance be as AP2 -f 2BAP. Join DA, DP, cutting the circle in E and T, and let the gravity be expounded by DA2, so that the gravity shall be to the resistance in P as DA2 to AP2+2BAP ; and the time of the whole ascent will be as the sector EDT of the circle. For draw DVQ,, cutting off the moment PQ, of the velocity AP, and the moment DTV of the sector DET answering to a given moment of time ; and that decrement PQ, of the velocity will be as the sum of the forces of gravity DA2 and of resistance AP2 + 2BAP, that is (by Prop. XII BookII,Elem.),asDP*. Then the arsa DPQ, which is proportional to PQ: is as DP2, and the area DTV, which is to the area DPQ, as DT2 to DP2, it as the given quantity DT2. Therefore the area EDT decreases uniformly according to the rate of the future time, by subduction of given particles DT V, and is therefore proportional to the time of the whole ascent. Q..E.D. CASE 2. If the velocity in the ascent of the body be expounded by the length AP as before, and the resistance be made as AP2 -f- 2BAP,and if the force of grav ity be less than can be expressed by DA2 ; take BD of such a length, that AB2 — BD2 maybe proportional to the gravity, and let DF be perpendicular and equal F O £S2 THE MATHEMATICAL PRINCIPLES [BOOK ll. to DB, and through the vertex F describe the hyperbola FTVE, whose con jugate semi -diameters are DB and DF; and which cuts DA in E, and DP, DQ in T and V ; and the time of the whole ascent will be as the hyper bolic sector TDE. For the decrement PQ of the velocity, produced in a given particle of time, is as the sum of the resistance AP2 -f 2BAP and of the gravity AB2 — BD2, that is, as BP2 — BD2. But the area DTV is to the area DPQ as DT2 to DP2 ; and, therefore, if GT be drawn perpendicular to DF. as GT2 or GD2 — DF2 to BD2, and as GD2 to BP2, and, by di vision, as DF2 to BP2 — BD2. Therefore since the area DPQ is as PQ, that is, as BP2 — BD2, the area DTV will be as the given quantity DF2. Therefore the area EDT decreases uniformly in each of the equal particles of time, by the subduction of so many given particles DTV, and therefore is proportional to the time. Q.E.D. CASE 3. Let AP be the velocity in the descent of """ the body, and AP2 + 2BAP the force of resistance, and BD2 — AB2 the force of gravity, the angle DBA being a right one. And if with the centre D, and the principal vertex B, there be described a rectangular hyperbola BETV cutting DA, DP, and DQ produced in E, T, and V : the sector DET of this hyperbola will D be as the whole time of descent. For the increment PQ of the velocity, and the area DPQ proportional to it, is as the excess of the gravity above the resistance, that is, as m)2?_ AB2 _2BAP — AP2 or BD2— BP2. And the area DTV is to the area DPQ as DT3 to DP2 ; and therefore as GT2 or GD" - BD2 to BP2, and as GD2 to BD2, and, by division, as BD2 to BD2 - BP2. Therefore since the ami DPQ is as BD2 — BP2, the area DTV will be as the given quantity BD2. Therefore the area EDT increases uniformly in the several equal particles of time by the addition of as many given particles DTV, and therefore is proportional to the time of the descent. Q.E.D. Con. If with the centre D and the semi-diameter DA there be drawn through the vertex A an arc A/ similar to the arc ET, and similarly sub- tendino^the angle A DT, the velocity AP will be to the velocity which the body in the time EDT, in a non-resisting space, can lose in its ascent, or acquire in its descent, as the area of the triangle DAP to the area of the Bector DA/ ; and therefore is given from the time given. For the velocity ir a non-resistin^ medium is proportional to the time, and therefore to this sector : in a resisting medium, it is as the triangle ; and in both mediums, where it is least, it approaches to the ratio of equality, as the sector and triangle do SEC. III.] OF NATURAL PHILOSOPHY. 283 SCHOLIUM. One may demonstrate also that case in the ascent of the body, where the force of gravity is less than can be expressed by DA2 or AB2 + BD2, and greater than can be expressed by AB2 — DB2, and must be expressed by AB2. But I hasten to other things PROPOSITION XIV. THEOREM XL The same things being supposed, 1 say, that the space described in the ascent or descent is as the difference of the area by which the time is expressed, and of some other area which is augmented or diminished in an arithmetical progression ; if the forces compounded of the re sistance and the gravity be taken, in a geometrical progression. Take AC (in these three figures) proportional to the gravity, and AK to the resistance ; but take them on the same side of the point A, if the \* "1 \ B A K QP body is descending, otherwise on the contrary. Erect A b, which make to DB as DB2 to 4BAC : and to the rectangular asymptotes CK, CH, de scribe the hyperbola 6N ; and, erecting KN perpendicular to CK, the area A/AK will be augmented or diminished in an arithmetical progression, while the forces CK are taken in a geometrical progression. I say, there fore, that the distance of the body from its greatest altitude is as the excess of the area A6NK above the area DET. For since AK is as the resistance, that is, as AP2 X 2BAP ; assume any given quantity Z, and put AK equal to then (by Lem, 284 THE MATHEMATICAL PRINCIPLES [BOOK II 2APQ, + 2B A X PU II of this Book) the moment KL of AK will be equal to 2BPQ or Z -, and the moment KLON of the area A£NK will be equal to 2BPQ X LO BPQ, X BD3 ~~Z~~ >r 2Z X OK x~AB" CASE 1. Now if the body ascends, and the gravity be as AB2 + BD9 BET being a circle, the line AC, which is proportional to the gravity AW2 i RF)2 will be - ~ -- , and DP2 or AP2 + 2BAP + AB2 + BD2 will be AK XZ + AC X Z or CK X Z : and therefore the area DTV will be to the area DPQ, as DT2 or I)B2 to CK X Z. CASE 2. If the body ascends, and the gravity be as AB2 — BD2, the A r>2 _ HI)2 line AC will be " -- ^ --- , and DT2 will be to DP2 as DF2 or DB2 Z to BP2— BD2 or AP2 + 2BAP + AB2 — BD2, that is, to AK X Z + H AC X Z or CK X Z. And therefore the area DTV will be to the area DPQ as DB2 to CK X Z. CASE 3. And by the same reasoning, if the body descends, and therefore the gravity is as BD2 — AB2, and the line AC becomes equal to or)2 AB2 5T— r ; the area DTV will be to the area DPQ as DB2 to CK X Z Z : as above. Since, therefore, these areas are always in this ratio, if for the area SEC. 111^ OF NATURAL PHILOSOPHY. 285 DTY, by which the moment of the time, always equal to itself, is express ed, there be put any determinate rectangle, as BD X m, the area DPQ,, that is, |BD X PQ, will be to BD X mas CK X Z to BD2. And thence PQ X BD3 becomes equal to 2 BD XmX CK X Z, and the moment KLON BP X BD X tn of the area A6NK, found before, becomes - .-^ — -- . From the area DET subduct its moment DTV or BD X m, and there will remain --- -Pp — — . Therefore the difference of the moments, that is, the AP X BD X m mo.nent of the difference of tne areas, is equal to -- 7-5 --- ; and therefore (because of the given quantity --- T-~ — ) as the velocity AP ; that is, as the moment of the space which the body describes in its ascent or descent. And therefore the difference of the areas, and that space, in creasing or decreasing by proportional moments, and beginning together or vanishing together, are proportional. Q.E.D. COR. If the length, which arises by applying the area DET to the line BD, be called M ; and another length V be taken in that ratio to the length M, which the line DA has to the line DE; the space which a body, in a resisting medium, describes in its whole ascent or descent, will be to the space which a body, in a non-resisting medium, falling from rest, can de scribe in the same time, as the difference of the aforesaid areas to BD X V2 - -TO"""" j an(^ therefore is given from the time given. For the space in a A.LJ non-resisting medium is in a duplicate ratio of the time, or as V2 ; and. BD X V2 because BD and AB are given, as --- -TTT- — . This area is equal to the DA2 X BD x M2 area -- fvGr*~~~T~R — "~ anc* ^ne moment Of M is m ; and therefore the DA2 X BD X 2M X m moment of this area is --- =- --- ^5 — -. But this moment is to "" X .A D the moment of the difference of the aforesaid areas DET and A6NK, viz., to AP X BD X m DA2 X BD X Mx DA2 . ^^ - -- , as - -r- - to |BD X AP, or as into DET to DAP ; and, therefore, when the areas DKT and DAP are least, in the BD X V2 ratio of equality. Therefore the area — r^ -- and the difference of the areas DET and A&NK, when all these areas are least, have equal moments ; and { re therefore equal. Therefore since the velocities, and therefore also the s] aces in both mediums described together, in the beginning of the de scent or the end of the ascent, approach to equality, and therefore are then 286 THE MATHEMATICAL PRINCIPLES [BOOK II BD X V2 one to another as the area r-^ — , and the difference of the areas DET AD and A6NK ; and moreover since the space, in a non-resisting medium, is BD X V2 perpetually as Tu~~> an(^ tne sPace> in a resisting medium, is perpetu ally as the difference of the areas DET and A&NK ; it necessarily follows, that the spaces, in both mediums, described in any equal times, are one to BD X V2 another as that area 7-5 — » an(^ ^he difference of the areas DET and A6NK. Q.E.D. SCHOLIUM. The resistance of spherical bodies in fluids arises partly from the tena city, partly from the attrition, and partly from the density of the medium. And that part of the resistance which arises from the density of the fluid is, as I said, in a duplicate ratio of the velocity ; the other part, which arises from the tenacity of the fluid, is uniform, or as the moment of the time ; and, therefore, we might now proceed to the motion of bodies, whicli are resisted partly by an uniform force, or in the ratio of the moments of the time, and partly in the duplicate ratio of the velocity. But it is suf ficient to have cleared the way to this speculation in Prop. VIII and IX foregoing, and their Corollaries. For in those Propositions, instead of the uniform resistance made to an ascending body arising from its gravity, one may substitute the uniform resistance which arises from the tenacity of the medium, when the body moves by its vis insita alone ; and when the body ascends in a right line, add this uniform resistance to the force of gravity, and subduct it when the body descends in a right line. One might also go on to the motion of bodies which are resisted in part uni formly, in part in the ratio of the velocity, and in part in the duplicate ratio of the same velocity. And I have opened a way to this in Prop. XIII and XIV foregoing, in which the uniform resistance arising from the tenacity of the medium may be substituted for the force of gravity, or be compounded with it as before. But I hasten to other things. SKC. -IV'.] OF NATUEAL PHILOSOPHY. 2S? SECTION IV. Of the circular motion of bodies in resisting' mediums. LEMMA III. Let PQR be a spiral rutting all the radii SP, SO, SR, <J*c., in equal angles. Draw tfie right line PT touching the spiral in any point P, and cutting the radius SQ in T ; cfo'er?0 PO, QO perpendicular to the spiral, and meeting- in, O, and join SO. .1 say, that if Hie points P a/*(/ Q approach and coincide, the angle PSO vri/Z become a right angle, and the ultimate ratio of the rectangle TQ, X 2PS to P^3 //>i// />e £/ie ya/io o/" equality. For from the right angles OPQ, OQR, sub duct the equal angles SPQ, SQR, and there will remain the equal angles OPS, OQS. Therefore a circle which passes through the points OSP will pass also through the point Q. Let the points P and Q, coincide, and this circle will touch the spiral in the place of coincidence PQ, and will therefore cut the right line OP perpendicularly. Therefore OP will become a diameter of this circle, and the angle OSP, being in a semi-circle, becomes a right one. Q.E.1). Draw Q,D, SE perpendicular to OP, and the ultimate ratios of the lines will be as follows : TO to PD as TS or PS to PE, or 2PO to 2PS • and PD to PO as PO to 2PO ; and, ex cequo pertorbatt, to TO to PO as PO to 2PS. Whence PO2 becomes equal to TO X 2PS. O.E.D. PROPOSITION XV. THEOREM XII. Tf the density of a medium in each place thereof be recipr on iJl y as the distance of the places from an immovable centre, aud the centripetal force be in the duplicate ratio of the density ; I say, that a body mny revolve in a spiral which cuts all the radii drawn from that centre in a given angle. Suppose every thing to be as in the forego ing Lemma, and produce SO to V so that SV may be equal to SP. In any time let a body, in a resisting medium, describe the least arc PO, and in double the time the least arc PR : and the decrements of those arcs arising from the resistance, or their differences from the arcs which would be described in a non-resist ing medium in the same times, will be to each other as the squares of the times in which they are generated ; therefore the decrement of the 288 THE MATHEMATICAL PRINCIPLES [_BoOK 11 arc PQ is the fourth part of the decrement of the arc PR. Whence also if the area QSr be taken equal to the area PSQ, the decrement of the arc PQ will be equal to half the lineola Rr ; and therefore the force of resist ance and the centripetal force are to each other as the lineola jRrandTQ which they generate in the same time. Because the centripetal force with which the body is urged in P is reciprocally as SP2, and (by Lem. X, Book I) the lineola TQ, which is generated by that force, is in a ratio compounded of the ratio of this force and the duplicate ratio of the time in which the arc PQ, is described (for in this case I neglect the resistance, as being infinitely less than the centripetal force), it follows that TQ X SP2, that is (by the last Lemma), fPQ2 X SP, will be in a duplicate ra tio of the time, and therefore the time is as PQ, X v/SP ; and the velo city of the body, with which the arc PQ is described in that time, as PQ 1 — -«p or , that is, in the subduplicate ratio of SP reciprocally. And, by a like reasoning, the velocity with which the arc QR is described, is in the subduplicate ratio of SQ reciprocally. Now those arcs PQ and QR are as the describing velocities to each other ; that is, in the subdu plicate ratio of SQ to SP, or as SQ to x/SP X SQ; and, because of the equal angles SPQ, SQ?', and the equal areas PSQ, QSr, the arc PQ is to the arc Qr as SQ to SP. Take the differences of the proportional conse quents, and the arc PQ will be to the arc Rr as SQ to SP — VSP X ~SQ~, or ^VQ. For the points P and Q coinciding, the ultimate ratio of SP — X SQ to |VQ is the ratio of equality. Because the decrement of the arc PQ arising from the resistance, or its double Rr, is as the resistance and the square of the time conjunctly, the resistance will be &Sp-^r — op. *1 X But PQ was to Rr as SQ to |VQ, and thence SSaTXToD becomes as Jr vst X oJr -VQ -OS pWxsvxSQ,' or ns ETp^TsP- For the poillts p and a coincidin& SP and SQ coincide also, and the angle PVQ becomes a right one; and, because of the similar triangles PVQ, PSO, PQ. becomes to '-VQ as OP OS to | OS. Therefore : y — — -- is as the resistance, that is, in the ratio of \J i X ol the density of the medium in P and the duplicate ratio of the velocity conjunc-tly. Subduct the duplicate ratio of the velocity, namely, the ratio 1 OS ^5, and there will remain the density of the medium in P. as 7^5 - = OA Ur X fei Let the spiral be given, and; because of the given ratio of OS to OP, the density of the medium in P will be as ~-p. Therefore in a medium whose SEC. IV,] OF NATURAL PHILOSOPHY. 2S9 density is reciprocally as SP the distance from the centre, a body will re volve in this spiral. Q.E.D. COR. 1. The velocity in any place P, is always the same wherewith a body in a non-resisting medium with the same centripetal force would re volve in a circle, at the same distance SP from the centre. COR. 2. The density of the medium, if the distance SP be given, is as OS OS TTp, but if that distance is not given, as ^ ^5. And thence a spiral may be fitted to any density of the medium. COR. 3. The force of the resistance in any place P is to the centripetal force in the same place as |OS to OP. For those forces are to each other ^VQ x PQ iPQ2 as iRr and TQ, or as 1—^-^~- and ^-, that is, as iVQ and PQ, ol%, ol or |OS and OP. The spiral therefore being given, there is given the pro portion of the resistance to the centripetal force ; and, vice versa, from that proportion given the spiral is given. COR. 4. Therefore the body cannot revolve in this spiral, except where the force of resistance is less than half the centripetal force. Let the re sistance be made equal to half the centripetal force, and the spiral will co incide with the right line PS, and in that right line the body will descend to the centre with a velocity that is to the velocity, with which it was proved before, in the case of the parabola (Theor. X, Book I), the descent would be made in a non-resisting medium, in the subduplicate ratio of unity to the number two. And the times of the descent will be here recip rocally as the velocities, and therefore given. COR. 5. And because at equal distances from the centre the velocity is the same in the spiral PQ,R as it is in the right line SP, and the length of the spiral is to the length of the right line PS in a given ratio, namely, in the ratio of OP to OS ; the time of the descent in the spiral will be to the time of the descent in the right line SP in the same given ratio, and therefore given. COR. 6. If from the centre S, with any two given intervals, two circles are described ; and these circles remaining, the angle which the spiral makes with the radius" PS be any how changed ; the number of revolutions which the body can complete in the space between the circumferences of those circles, going PS round in the spiral from one circumference to another, will be as 7^, or as Ok5' ths tangent of the angle which the spiral makes with the radius PS ; and 19 290 THE MATHEMATICAL PRINCIPLES [BOOK II OP the time of the same revolutions will be as -^, that is, as the secant of the Uo same angle, or reciprocally as the density of the medium. COR. 7. If a body, in a medium whose density is reciprocally as the dis tances of places from the centre, revolves in any curve AEB about that centre, and cuts the first radius AS in the same angle in B as it did before in A, and that with a velocity that shall be to its first velocity in A re ciprocally in a subduplicate ratio of the distances from the centre (that is, as AS to a mean propor tional between AS and BS) that body will con tinue to describe innumerable similar revolution? BFC, CGD, &c., and by its intersections will distinguish the radius AS into parts AS, BS, CS, DS, &c., that are con tinually proportional. But the times of the revolutions will be as the perimeters of the orbits AEB, BFC, CGD, &c., directly, and the velocities 3 3 at the beginnings A, B, C of those orbits inversely ; that is as AS % BS % CS"2". And the whole time in which the body will arrive at the centre, will be to the time of the first revolution as the sum of all the continued 142 proportionals AS2, BS2, CS2, going on ad itifinitum, to the first term * i 3 AS2 ; that is, as the first term AS2 to the difference of the two first AS2 — BS% or as f AS to AB very nearly. Whence the whole time may be easily found. COR. 8. From hence also may be deduced, near enough, the motions of bodies in mediums whose density is either uniform, or observes any other assigned law. From the centre S, with intervals SA, SB, SC, &c., con tinually proportional, describe as many circles ; and suppose the time of the revolutions between the perimeters of any two of those circles, in the medium whereof we treated, to be to the time of the revolutions between the same in the medium proposed as the mean density of the proposed me dium between those circles to the mean density of the medium whereof we treated, between the same circles, nearly : and that the secant of the angle in which the spiral above determined, in the medium whereof we treated, cuts the radius AS, is in the same ratio to the secant of the angle in which the new spiral, in the proposed medium, cuts the same radius : and also that the number of all the revolutions between the same two circles is nearly as the tangents of those angles. If this be done every where between every two circles, the motion will be continued through all the circles. And by this means one may without difficulty conceive at what rate and in what time bodies ought to revolve in any regular medium. •SEC. IY.1 OF NATURAL PHILOSOPHY. 291 COR. 9. And although these motions becoming eccentrical should be performed in spirals approaching to an oval figure, yet, conceiving the several revolutions of those spirals to be at the same distances from each other, and to approach to the centre by the same degrees as the spiral above described, we may also understand how the motions of bodies may be per formed in spirals of that kind. PROPOSITION XVI. THEOREM XIII. If the density of the medium in each of the places be reciprocally as the distance of the >, places from the immoveable centre, and the centripetal force be reciprocally as any power of the same distance, I say, that the body may revolve in a spiral intersecting all the radii drawn from that centre in a given, angle. This is demonstrated in the same manner as the foregoing Proposition. For if the centri petal force in P be reciprocally as any power SPn+1 of the distance SP whose index is n + 1 ; it will be collected, as above, that the time in which the body describes any arc PQ, i will be as PQ, X PS2U ; and the resistance in i!! x _ n;°raS "~' X SPn;PQ, X SP"XSQ,' , , 1 — in X OS . 1 — \n X OS . therefore as Qp"^~gpirqTT' tliat 1S> (because - ~~Qp~~ 1S a £lven quantity), reciprocally as SPn+ ! . And therefore, since the velocity is recip rocally as SP3", the density in P will be reciprocally as SP. COR. 1. The resistance is to the centripetal force as 1 — ^//. X OS to OP. COR. 2. If the centripetal force be reciprocally as SP3. 1 — ±w will be === 0 ; and therefore the resistance and density of the medium will be nothing, as in Prop. IX, Book I. COR. 3. If the centripetal force be reciprocally as any power of the ra dius SP, whose index is greater than the number 3, the affirmative resist ance will be changed into a negative. SCHOLIUM. This Proposition and the former, which relate to mediums of unequal density, are to be understood of the motion of bodies that are so small, that the greater density of the medium on one side of the body above that on the other is not to be considered. I suppose also the resistance, cateris paribus, to be proportional to its density. Whence, in mediums whose 292 THE MATHEMATICAL PRINCIPLES | BoOK II force of resistance is not as the density, the density must be so much aug mented or diminished, that either the excess of the resistance may be taken away, or the defect supplied. PROPOSITION XVII. PROBLEM IV To find the centripetal force and the resisting force of the medium, by which a body, the law of the velocity being given, shall revolve in a given spiral. Let that spiral be PQR. From the velocity, with which the body goes over the very small arc PQ,, the time will be given : and from the altitude TQ,, which is as the centripetal force, and the square of the time, that force will be given. Then from the difference RSr of the areas PSQ, and Q,SR described in equal particles of time, the re tardation of the body will be given ; and from the retardation will be found the resisting force and density of the medium. PROPOSITION XVIII. PROBLEM V. The law of centripptal force being given, to find the density of the me dium in each of the places thereof, by which a body may describe a given spiral. From the centripetal force the velocity in each place must be found ; then from the retardation of the velocity the density of the medium is found, as in the foregoing Proposition. But I have explained the method of managing these Problems in the tenth Proposition and second Lemma of this Book; and will no longer detain the reader in these perplexed disquisitions. I shall now add some things relating to the forces of progressive bodies, and to the density and resistance of those mediums in which the motions hitherto treated of, and those akin to them, are performed. SEC. V.] OF NATURAL PHILOSOPHY. 293 SECTION V. Of the density and compression of fluids ; and of hydrostatics. THE DEFINITION OF A FLUID. A fluid is any body whose parts yield to any force impressed on it, by yielding, are easily moved among themselves. PROPOSITION XIX. THEOREM XIv All the parts of a homogeneous and unmoved fluid included in any un moved vessel, and compressed on every side (setting aside the consider ation of condensation, gravity, and all centripetal forces], will be equally pressed on every side, and remain in their places without any motion arising from that pressure. CASE 1. Let a fluid be included in the spherical vessel ABC, and uniformly compressed on every side : 1 say, that no part of it will be moved by that pressure. For if any part, as L), be moved, all such parts at the same distance from the centre on every side must necessarily be moved at the same time by a like motion ; because the pressure of them all is similar and equal ; and all other motion is excluded that does not arise from that pressure. But if these parts come all of them nearer to the centre, the fluid must be condensed towards the centre, contrary to the supposition. If they recede from it, the fluid must be condensed towards the circumfer ence ; which is also contrary to the supposition. Neither can they move in any one direction retaining their distance from the centre, because for the same reason, they may move in a contrary direction : but the sami part cannot be moved contrary ways at the same time. Therefore no part of the fluid will be moved from its place. Q,.E.T). CASE 2. I say now, that all the spherical parts of this fluid are equally pressed on every side. For let EF be a spherical part of the fluid ; if this be not pressed equally on every side, augment the lesser pressure till it be pressed equally on every side; and its parts (by Case I) will remain in their places. But before the increase of the pressure, they would remain in their places (by Case 1) ; and by the addition of a new pressure they will be moved, by the definition of a fluid, from those places. Now these two conclusions contradict each other. Therefore it was false to say that the sphere EF was not pressed equally on every side. Q,.E.D. CASE 3. I say besides, that different spherical parts have equal pressures. For the contiguous spherical parts press each other mutually and equally in the point of contact (by Law III). But (by Case 2) they are pressed on every side with the same force. Therefore any two spherical parts lot 391 THE MATHEMATICAL PRINCIPLES [BoOK II. contiguous, since an intermediate spherical part can touch both, will be pressed with the same force. Q.E.D. CASE 4. I say now, that all the parts of the fluid are every where press ed equally. For any two parts may be touched by spherical parts in any points whatever ; and there they will equally .press those spherical parts (by Case 3). and are reciprocally equally pressed by them (by Law III). Q.E.D. CASE 5. Since, therefore, any part GHI of the fluid is inclosed by the rest of the fluid as in a vessel, and is equally pressed on every side ; and also its parts equally press one another, and are at rest among themselves ; it is manifest that all the parts of any fluid as GHI, which is pressed equally on every side, do press each other mutually and equally, and are at rest among themselves. Q.E.D. CASE 6. Therefore if that fluid be included in a vessel of a yielding substance, or that is not rigid, and be not equally pressed on every side, the same will give way to a stronger pressure, by the Definition of fluidity. CASE 7. And therefore, in an inflexible or rigid vessel, a fluid will not Sustain a stronger pressure on one side than on the other, but will give way to it, and that in a moment of time ; because the rigid side of the vessel does not follow the yielding liquor. But the fluid, by thus yielding, will press against the opposite side, and so the pressure will tend on every side to equality. And because the fluid, as soon as it endeavours to recede from the part that is most pressed, is withstood by the resistance of the vessel on the opposite side, the pressure will on every side be reduced to equality, in a moment of time, without any local motion : and from thence the parts of the fluid (by Case 5) will press each other mutually and equal ly, and be at rest among themselves. Q..E.D. COR. Whence neither will a motion of the parts of the fluid among themselves be changed by a pressure communicated to the external super ficies, except so far as either the figure of the superficies may be somewhere altered, or that all the parts of the fluid, by pressing one another more in tensely or remissly, may slide with more or less difficulty among them selves. PROPOSITION XX. THEOREM XV. Jf all the parts of a spherical fluid, homogeneous at equal distances from the centre, lying on. a spherical concentric bottom, gravitate towards the centre of the whole, the bottom will sustain the weight of a cylin der, whose base is equal to the superficies of the bottom, and whose al titude is the same with that of the incumbent fluid. Let OHM be the superficies of the bottom, and AEI the upper super ficies of the fluid. Let the fluid be distinguished into concentric orbs of 3qual thickness, by the innumerable spherical superficies *3PK, CGL : and SEC. V OF NATURAL PHILOSOPHY. 295 conceive the force of gravity to act only in the upper superficies of every orb, and the actions to be equal on the equal parts of all the su perficies. Therefore the upper superficies AE is pressed by the single force of its own grav ity, by which all the parts of the upper orb, and the second superficies BFK, will (by Prop. XIX), according to its measure, be equally pressed. The second superficies BFK is pressed likewise by the force of its own gravity, which, added to the former force, makes the pressure double. The third superficies CGL is, according to its measure, acted on by this pressure and the force of its own gravity besides, which makes its pressure triple. And in like manner the fourth superfi cies receives a quadruple pressure, the fifth superficies a quintuple, and so on. Therefore the pressure acting on every superficies is not as the solid quantity of the incumbent fluid, but as the number of the orbs reaching to the upper surface of the fluid ; and is equal to the gravity of the lowest orb multiplied by the number of orbs : that is, to the gravity of a solid whose ultimate ratio to the cylinder above-mentioned (when the number of the orbs is increased and their thickness diminished, ad infiititum, so that the action of gravity from the lowest superficies to the uppermost may be- some continued) is the ratio of equality. Therefore the lowest superficies •sustains the weight of the cylinder above determined. Q..E.D. And by a like reasoning the Proposition will be evident, where the gravity of the fluid decreases in any assigned ratio of the distance from the centre, and also where the fluid is more rare above and denser below. Q.E.D. COR. 1. Therefore the bottom is not pressed by the whole weight of the incumbent fluid, but only sustains that part of it which is described in the Proposition ; the rest of the weight being sustained archwise by the spheri cal figure of the fluid. COR. 2. The quantity of the pressure is the same always at equal dis tances from the centre, whether the superficies pressed be parallel to the horizon, or perpendicular, or oblique ; or whether the fluid, continued up wards from the compressed superficies, rises perpendicularly in a rectilinear direction, or creeps obliquely through crooked cavities and canals, whether those passages be regular or irregular, wide or narrow. That the pressure is not altered by any of these circumstances, may be collected by applying the demonstration of this Theorem to the several cases of fluids. COR. 3. From the same demonstration it may also be collected (by Prop. XIX), that the parts of a heavy fluid acquire no motion among themselvei by the pressure of the incumbent veight, except that motion which arises from condensation. 296 THE MATHEMATICAL PRINCIPLES [BCOK II Con. 4. And therefore if another body of the same specific gravity, in capable of condensation, be immersed in this fluid, it will acquire no mo tion by the pressure of the incumbent weight: it will neither descend nor . ascend, nor change its figure. If it be spherical, it will remain so, notwith standing the pressure ; if it be square, it will remain square; and that, whether it be soft or fluid : whether it swims freely in the fluid, or lies at the bottom. For any internal part of a fluid is in the same state with the submersed body ; and the case of all submersed bodies that have the same magnitude, figure, and specific gravity, is alike. If a submersed body, re taining its weight, should dissolve and put on the form of a fluid, this body, if before it would have ascended, descended, or from any pressure as sume a new figure, would now likewise ascend, descend, or put on a new figure ; and that, because its gravity and the other causes of its motion remain. But (by Case 5, Prop. XIX; it would now be at rest, and retain its figure. Therefore also in the former case. COR. 5. Therefore a body that is specifically heavier than a fluid con tiguous to it will sink ; and that which is specifically lighter will ascend, and attain so much motion and change of figure as that excess or defect of gravity is able to produce. For that excess or defect is the same thing as an impulse, by which a body, otherwise in equilibria with the parts of the fluid, is acted on: and may be compared with the excess or defect of a weight in one of the scales of a balance. COR. 6. Therefore bodies placed in fluids have a twofold gravity • the one true and absolute, the other apparent, vulgar, and comparative. Ab solute gravity is the whole force with which the body tends downwards ; relative and vulgar gravity is the excess of gravity with which the body tends downwards more than the ambient fluid. By the first kind of grav ity the parts of all fluids and bodies gravitate in their proper places ; and therefore their weights taken together compose the weight of the whole. For the whole taken together is heavy, as may be experienced in vessels full of liquor ; and the weight of the whole is equal to the weights of all the parts, and is therefore composed of them. By the other kind of grav ity bodies do not gravitate in their places ; that is, compared with one another, they do not preponderate, but, hindering one another's endeavours to descend, remain in their proper places, as if they were not heavy. Those things which are in the air, and do not preponderate, are commonly looked on as not heavy. Those which do preponderate are commonly reckoned heavy, in as much as they are not sustained by the weight of the air. The Common weights are nothing else but the excess of the true weights above the weight of the air. Hence also, vulgarly, those things are called light which are less heavy, and, by yielding to the preponderating air, mount upwards. But these are only comparatively lig s &mA not truly so, because hey descend in racuo. Thus, in water, bodies *>icfc. by their greater or SEC. V.] OF NATURAL PHILOSOPHY. 297 less gravity, descend or ascend, are comparatively and apparently heavy or light ; and their comparative and apparent gravity or levity is the excess .or defect by which their true gravity either exceeds the gravity of the water or is exceeded by it. But those things which neither by preponder ating descend, nor, by yielding to the preponderating fluid, ascend, although by their true weight they do increase the weight of the whole, yet com paratively, and in the sense of the vulgar, they do not gravitate in the wa ter. For these cases are alike demonstrated. COR. 7. These things which have been demonstrated concerning gravity take place in any other centripetal forces. COR. 8. Therefore if the medium in which any body moves be acted on either by its own gravity, or by any other centripetal force, and the body be urged more powerfully by the same force ; the difference of the forces is that very motive force, which, in the foregoing Propositions, I have con sidered as a centripetal force. But if the body be more lightly urged by that force, the difference of the forces becomes a centrifugal force, and is tc be considered as such. COR. 9. But since fluids by pressing the included bodies do not change their external figures, it appears also (by Cor. Prop. XIX) that they will not change the situation of their internal parts in relation to onf another ; and therefore if animals were immersed therein, and that all sen sation did arise from the motion of their parts, the fluid will neither hurt the immersed bodies, nor excite any sensation, unless so far as those bodies may be condensed by the compression. And the case is the same of any system of bodies encompassed with a compressing fluid. All the parts of the system will be agitated with the same motions as if they were placed in a vacuum, and would only retain their comparative gravity ; unless so far as the fluid may somewhat resist their motions, or be requisite to con- glutinate them by compression. PROPOSITION XXI. THEOREM XVI. £<et the density of any fluid be proportional to the compression, and its parts be attracted downwards by a centripetal force reciprocally pro portional to the distances from the centre : I say, that, if those dis tances be taken continually proportional, the densities of the fluid at the same distances will be also continually proportional. Let ATV denote the spherical bottom of the fluid, S the centre, S A, SB. SC, SD, SE, SF, &c., distances continually proportional. Erect the per pendiculars AH, BI, CK, DL, EM, PN, &c., which shall be as the densi ties of the medium in the places A, B, C, D, E, F : and the specific grav ATT RT f^K" ities in those places will be aa -r-, , - &c., or, which is all one, a&- 298 THE MATHEMATICAL PRINCIPLES [BOOK II. AH BI CK ATT BC' CD' Suppose, first, these gravities to be uniformly continued from A to B, from B to C, from C to D, (fee., the decrements in the points B, C, D, (fee., being taken by steps. Arid these gravi ties drawn into the altitudes AB, BC, CD, (fee., will give the pressures AH, BI, CK, (fee., by which the bot tom ATV is acted on (by Theor. XV). Therefore the particle A sustains all the pressures AH, BI, CK, DJL, (fee., proceeding in infinitum ; and the particle B sus tains the pressures of all but the first AH ; and the par ticle C all but the two first AH, BI ; and so on : and therefore the density AH of the first particle A is to the density BI of the second particle B as the sum of all AH -f- BI + CK + DL, in infinitum, to the sum of all BI -f- CK -f DL, (fee. And BI the density of the second particle B is to CK the density of the third C, as the sum of all BI -f CK + DL, (fee., to the sum of all CK -f- DL, (fee. Therefore these sums are proportional to their differences AH, BI, CK, (fee., and therefore continually propor tional (by Lem. 1 of this Book) ; and therefore the differences AH, BI, CK, (fee., proportional to the sums, are also continually proportional. Wherefore since the densities in the places A, B, C, (fee., are as AH, BI, CK, (fee., they will also be continually proportional. Proceed intermis- sively, and, ex ccquo, at the distances SA, SC, SE, continially proportional, the densities AH, CK, EM will be continually proportional. And by the same reasoning, at any distances SA, SD, SG, continually proportional, the densities AH. DL, GO, will be continually proportional. Let now the points A, B, C. D, E, (fee., coincide, so that the progression of the specif'.c gravities from the bottom A to the top of the fluid may be made continual ; and at any distances SA, SD, SG, continually proportional, the densities AH, DL, GO, being all along continually proportional, will still remain continually proportional. Q.E.D. COR. Hence if the density of the fluid in two places, as A and E, be given, its density in any other place Q, may be collected. With the centre S, and the rectan gular asymptotes SQ, SX, describe an hyperbola cut ting the perpendiculars AH, EM, QT in «, e, and q} as also the perpendiculars HX, MY, TZ, let fall upon the asypmtote SX, in //, 'm, and t. Make the area Y////Z to the given area YmAX as the given area EeqQ to the given area EeaA ; and the line Z£ produced will cut off the line Q,T. proportional to the density. For if the lines SA, SE, SQ are continually proportional, the areas ReqQ., fyaA will be equal, and thence X SEC. V. OF NATURAL PHILOSOPHY. 299 the areas YwYZ. X/zwY, proportional to them, will be also equal ; and the lines SX, SY, SZ, that is, AH, EM, Q,T continually proportional, as they ought to be. And if the lines SA, SE, SQ,5 obtain any other order in the series of continued proportionals, the lines AH, EM, Q,T, because of the proportional hyperbolic areas, will obtain the same order in another series of quantities continually proportional. PROPOSITION XXII THEOREM XVII. Let the density of any fluid be proportional to the compression, and its parts be attracted downwards by a gravitation reciprocally propor tional to the squares of the distances from the centre : I say, that if the distances be taken in harmonic progression, the densities of the fluid at those distances will be in a geometrical progression. Let S denote the centre, and SA, SB, SC, SD, SE, the distances in geometrical progression. Erect the perpendiculars AH, BI, CK, (fee., which shall be as the densities of c the fluid in the places A, B, C, D, » E, (fee., and the specific gravities thereof in those places will be as AH BI ,^-, (fee. Suppose these V LN M I. K i V V V V7 ^ n w r / £ L / / /t SA2'SB2'SC2' gravities to be uniformly continued, the first from A to B, the second from B to C, the third from C to I), &c. And these drawn into the altitudes AB, BC, CO, DE, (fec.; or, which is the same thing/into the distances SA, ATT r>T OT7" SB, SC, (fee., proportional to those altitudes, will give -~-r-, ^=5, -~~, (fee.. the exponents of the pressures. Therefore since the densities are as th^ sums of those pressures, the differences AH — BI, BI — CK, (fee., of tb,« densities will be as the differences of those sums ~-r~, ^, ~~, (fee. With the centre S, and the asymptotes SA, S#, describe any hyperbola, cutting the perpendiculars AH, BI, CK, (fee., in a, 6, c, (fee., and the perpendicu lars H/, I//,, K?#, let fall upon the asymptote Sv, in h, i, k ; and the dif ferences of the densities tu, uw, (fee., will be as „ A , ^^, (fee. And the SA; SB; rectangles tu X th, uw X uij (fee., or tp, uq, (fee., as that is, as Aa, Bb, (fee. AH X th BI X ui — , (fee. SA SB For, by the nature of the hyperbola, SA is to AH or St as th to Ar, and therefore — pri — is equal to Aa . And, by a like SA 300 THE MATHEMATICAL PRINC. PLES [BOOK II. reasoning, — ^n~~ *s e(lua^ to ^, &c- But Aa> B^> ^c, &cv are continu ally proportional, and therefore proportional to their differences Aa — B&, B6 — Cc; &c., therefore the rectangles fy?, nq, &c., are proportional to those differences ; as also the sums of the rectangles tp + uq, or tp + uq -f w to the sums of the differences Aa — Cc or Aa — Da7. Suppose several of these terms, and the sum of all the differences, as Aa — F/, will be pro portional t? the sum of all the rectangles, as zthn. Increase the number of terms, and diminish the distances of the points A, B, C, (fee., in iiijini- tum, and those rectangles will become equal to the hyperbolic area zthn. and therefore the difference Aa — F/ 19 proportional to this area. Take nowT any distances, as SA, SD, SF, in harmonic progression, and the dif ferences Aa — Da7, Da1 — F/ will be equal ; and therefore the areas thlx, xlnz, proportional to those differences will be equal among themselves, and the densities St, S:r, Sz, that is, AH, DL, FN, continually proportional. Q.E.D. COR. Hence if any two densities of the fluid, as AH and BI, be given, the area thiu, answering to their difference tu, will be given; and thence the density FN will be found at any height SF, by taking the area thnz to that given area thiu as the difference Aa — F/ to the difference Aa — Eh. SCHOLIUM. By a like reasoning it may be proved, that if the gravity of the particles of a fluid be diminished in a triplicate ratio of the distances from the centre ; and the reciprocals of the squares of the distances SA, SB, SC, &c., (namely, SA3 SA3 SA3 . opt' ^e ta^en m an arithmetical progression, the densities AH. BI, CK, &c., will be in a geometrical progression. And if the gravity be diminished in a quadruplicate ratio of the distances, and the reciprocals of the cubes of the distances (as ^-r^, SRS» sps' ^c'^ ^e ta^cn ^n ai>itnmeti- cai progression, the densities AH, BI, CK, &c., will be in geometrical pro gression. And so in irtfinitum. Again ; if the gravity of the particles of the fluid be the same at all distances, and the distances be in arithmetical progression, the densities will be in a geometrical progression as Dr. Hal- ley has found. If the gravity be as the distance, and the squares of the distances be in arithmetical progression, the densities will be in geometri cal progression. And so in infinitum. These things will be so, when the density of the fluid condensed by compression is as the force of compres sion ; or, which is the same thing, when the space possessed by the fluid is reciprocally as this force. Other laws of condensation may be supposed, as that the cube of the compressing force may be as the biquadrate of the SEC. V.] OF NATURAL PHILOSOPHY. 301 de isity ; or the triplicate ratio of tlie force the same with the quadruplicate ratio of the density : in which case, if the gravity he reciprocally as the square of the distance from the centre, the density will be reciprocally at the cube of the distance. Suppose that the cube of the compressing force be as the quadrato-cube of the density ; and if the gravity be reciprocally as the square of the distance, the density will be reciprocally in a sesqui- plicate ratio of the distance. Suppose the compressing force to be in a du plicate ratio of the density, and the gravity reciprocally in a duplicate ra tio of the distance, and the density will be reciprocally as 'the distance. To run over all the cases that might bo offered would be tedious. But as to our own air, this is certain from experiment, that its density is either accurately, or very nearly at least, as the compressing force ; and therefore the density of the air in the atmosphere of the earth is as the weight of the whole incumbent air, that is, as the height of the mercury in the ba rometer. PROPOSITION XXIII. THEOREM XVIII. If a fluid be composed of particles mutually flying each other, and the drnsity be as the compression, the centrifugal forces of the particles 'will be reciprocally proportional to tlie distances of their centres. And, vice versa, particles flying each otli,er, with forces that are reciprocally proportional to the distances of their centres^ compose an elastic fluid, whose density is as the compression. Let the fluid be supposed to be included in a cubic space ACE, and then to be reduced by compression into a lesser cubic space ace ; and the distances of the par- F tides retaining a like situation with respect to each other in both the spaces, will be as the sides AB, ab of the cubes ; and the densities of the mediums will be re ciprocally as the containing spaces AB3, ab3. In the plane side of the greater cube A BCD take the square DP equal to the plane side db of the lesser cube: and, by the supposition, the pressure with which the square DP urges the inclosed fluid will be to the pressure with which that square db urges the inclosed fluid as the densities of the me diums are to each other, that is, asa/>3 to AB3. But the pressure with which the square DB urges the included fluid is to the pressure with which the square DP urges the same fluid as the square DB to the square DP, that is, as AB2 to abz. Therefore, ex cequo, the pressure with which the square DB urges the fluid is to the pressure with which the square db urges the fluid as ab to AB. Let the planes FGH,/°V?, U drawn through the middles of the two cubes, and divide the fluid into tw^/ parts, These parts will press each other mutually with the same forces with which they A THE MATHEMATICAL PRINCIPLES [BOOK II. are themselves pressed by the planes AC, ac, that is, in the proportion of ab to AB : arid therefore the centrifugal forces by which these pressures are sustained are in the same ratio. The number of the particles being equal, and the situation alike, in both cubes, the forces which all the par ticles exert, according to the planes FGH,/o7/,, upon all, are as the forces which each exerts on each. Therefore the forces which each exerts on each, according to the plane FGH in the greater cube, are to the forces which each exerts on each, according to the plane fgh in the lesser cube, us ab to AB,*that is, reciprocally as the distances of the particles from each other. Q.E.D. And, vice versa, if the forces of the single particles are reciprocally as the distances, that is, reciprocally as the sides of the cubes AB, ab ; the sums of the forces will be in the same ratio, and the pressures of the sides i)B, db as the sums of the forces ; and the pressure of the square DP to the pressure of the side DB as ab2 to AB 2. And, ex cequo, the pressure of the square DP to the pressure of the side db as ab* to AB3 ; that is, the force of compression in the one to the force of compression in the other as the density in the former to the density in the latter. Q.E.D. SCHOLIUM. By a like reasoning, if the centrifugal forces of the particles are recip rocally in the duplicate ratio of the distances between the centres, the cubes of the compressing forces will be as the biquadrates of the densities. If the centrifugal forces be reciprocally in the triplicate or quadruplicate ratio of the distances, the cubes of the compressing forces will be as the quadrato- cubes, or cubo-cubes of the densities. And universally, if D be put for the distance, and E for the density of the compressed fluid, and the centrifugal forces be reciprocally as any power Dn of the distance, whose index is the number ??, the compressing forces will be as the cube roots of the power En + 2. whose index is the number n + 2 ; and the contrary. All these things are to be understood of particles whose centrifugal forces terminate in those particles that are next them, or are diffused not much further. We have an example of this in magnetical bodies. Their attractive vir tue is terminated nearly in bodies of their own kind that are next them. The virtue of the magnet is contracted by the interposition of an iron plate, and is almost terminated at it : for bodies further off are not attracted by the magnet so much as by the iron plate. If in this manner particles repel others of their own kind that lie next them, but do not exert their virtue on the more remote, particles of this kind will compose such fluids as are treated of in this Proposition, If the virtue of any particle diffuse itself every way in inftnitum, there will be required a greater force to produce an equal condensation of a greater quantity of the flui 1. But whether SEC. VI.] OF NATURAL PHILOSOPHY. 303 elastic fluids do really consist of particles so repelling each other, is a phy sical question. We have here demonstrated mathematically the property of fluids consisting of particles of this kind, that hence philosophers may take occasion to discuss that question. SECTION VI. Of the motion and resistance of funependulous bodies. PROPOSITION XXIV. THEOREM XIX. The quantities of matter i/i funependulous bodies, whose centres of oscil lation are equally distant from, the centre of suspension, are in a, ratio compounded of the ratio of the weights and the duplicate ratio of the times of the oscillations in vacuo. For the velocity which a given force can generate in a given matter in a given time is as the force and the time directly, and the matter inversely. The greater the force or the time is, or the less the matter, the greater ve locity will he generated. This is manifest from the second Law of Mo tion. Now if pendulums are of the same length, the motive forces in places equally distant from the perpendicular are as the weights : and therefore if two bodies by oscillating describe equal arcs, and those arcs are divided into equal parts ; since the times in which the bodies describe each of the correspondent parts of the arcs are as the times of the whole oscillations, the velocities in the correspondent parts of the oscillations will be to each other as the motive forces and the whole times of the oscillations directly, and the quantities of matter reciprocally : and therefore the quantities of matter are as the forces and the times of the oscillations directly and the velocities reciprocally. But the velocities reciprocally are as the times, and therefore the times directly and the velocities reciprocally are as the squares of the times; and therefore the quantities of matter are as the mo tive forces and the squares of the times, that is, as the weights and the squares of the times. Q.E.D. COR. 1. Therefore if the times are equal, the quantities of matter in each of the bodies are as the weights. COR. 2. If the weights are equal, the quantities of matter will be as the pquarcs of the times. COR. 3. If the quantities of matter are equal, the weights will be recip rocally as the squares of the times. COR. 4. Whence since the squares of the times, cceteris paribus, are as the length* of the pendulums, therefore if both the times and quantities of matter are equal, the weights will be as the lengths of the pendulums. J04 THE MATHEMATICAL PRINCIPLES [BOOK 11 COR. 5. And universally, the quantity of matter in the pendulous body is as the weight and the square of the time directly, and the length of the pendulum inversely. COR. 6. But in a non-resisting medium, the quantity of matter in the pendulous body is as the comparative weight and the square of the time directly, and the length of the pendulum inversely. For the comparative weight is the motive force of the body in any heavy medium, as was shewn above ; and therefore does the same thing in such a non-resisting medium as the absolute weight does in a vacuum. COR. 7. And hence appears a method both of comparing bodies one among another, as to the quantity of matter in each ; and of comparing the weights of the same body in different places, to know the variation of its gravity. And by experiments made with the greatest accuracy, I have always found the quantity of matter in bodies to be proportional to their weight. PROPOSITION XXV. THEOREM XX. Funependulous bodies that are, in, any 'medium, resisted in the ratio oj the moments of time, and funepetidulons bodies that move in a non- resisting medium of the same specific gravity, perform their oscilla tions in. a cycloid in the same time, and describe proportional parts oj arcs together. Let AB be an arc of a cycloid, which a body D, by vibrating in a non-re sisting medium, shall describe in any time. Bisect that arc in C, so that C may be the lowest point thereof ; and the accelerative force with which the body is urged in any place D, or d or E, will be as the length of the arc CD, pressed by that same arc ; and since the resistance is as the moment of the time, and therefore given, let it ba expressed by the given part CO of the cycloidal arc, and take the arc Od in the same ratio to the arc CD that the arc OB has to the arc CB : and the force with which the body in d is urged in a resisting medium, being the excess of the force Cd above the resistance CO, will be expressed by the arc Od, and will therefore be to the force with which the body D is urged in a non-resisting medium in the place D, as the arc Od to the arc CD ; and therefore also in the place B, as the arc OB to the arc CB. Therefore if two bodies D, d go from the place B, and are urged by these forces ; since the forces at the beginning are as the arc CB and OB, the first velocities and arcs first described will be in the same ratio. Let those arcs be BD and Ed, and the remaining arcf SEC. VI. | OF NATURAL PHILOSOPHY. 305 CD, Odj will be in the same ratio. Therefore the forces, being propor tional to those arcs CD, Od, will remain in the same ratio as at the be ginning, and therefore the bodies will continue describing together arcs in the same ratio. Therefore the forces and velocities and the remaining arcs CD. Od, will be always as the whole arcs CB, OB, and therefore those re maining arcs wLl be described together. Therefore the two bodies D and d will arrive together at the places C and O ; that whicli moves in the non-resisting medium, at the place C, and the other, in the resisting me dium, at the place O. Now since the velocities in C and O are as the arcs CB, OB, the arcs which the bodies describe when they go farther will be in the same ratio. Let those arcs be CE and Oe. The force with which the body D in a non-resisting medium is retarded in E is as CE, and the force with which the body d in the resisting medium is retarded in e, is as the sum of the force Ce and the resistance CO, that is, as Oe ; and there fore the forces with which the bodies are retarded are as the arcs CB, OB, proportional to the arcs CE, Oe ; and therefore the velocities, retarded in that given ratio, remain in the same given ratio. Therefore the velocities and the arcs described with those velocities are always to each other in that given ratio of the arcs CB and OB ; and therefore if the entire arcs AB, aB are taken in the same ratio, the bodies D andc/ will describe those aics together, and in the places A and a will lose all their motion together. Therefore the whole oscillations are isochronal, or are performed in equal times ; and any parts of the arcs, as BD, Ed, or BE, Be, that are described together, are proportional to the whole arcs BA, B«. Q,.E.D. COR. Therefore the swiftest motion in a resisting medium does not fall upon the lowest point C, but is found in that point O, in which the whole arc described Ba is bisected. And the body, proceeding from thence to a, is retarded at the same rate with which it was accelerated before in its de scent from B to O. PROPOSITION XXVI. THEOREM XXI. Funependulous bodies, that are resisted in the ratio of the velocity, have their oscillations in a cycloid isochronal. For if two bodies, equally distant from their centres of suspension, de scribe, in oscillating, unequal arcs, and the velocities in the correspondent parts of the arcs be to each other as the whole arcs ; the resistances, pro portional to the velocities, will be also to each other as the same arcs. Therefore if these resistances be subducted from or added to the motive forces arising from gravity which are as the same arcs, the differences or sums will be to each other in the same ratio of the arcs ; and since the in crements and decrements of the velocities are as these differences or sums, the velocities will be always as the whole arcs; therefore if the velocities are in any one case as the whole arcs, they will remain always in the same 20 306 THE MATHEMATICAL PRINCIPLES [BOOK. 11 ratio. But at the beginning of the motion, when the bodies begin to de scend and describe those arcs, the forces, which at that time are proportional to the arcs, will generate velocities proportional to the arcs. Therefore the velocities will be always as the whole arcs to be described, and there fore those arcs will be described in the same time. Q,.E.D. PROPOSITION XXVII. THEOREM XXII. If fnnependulous bodies are resisted in the duplicate ratio of their velocities, the differences between the times of the oscillations in a re sisting medium, and the times of the oscillations in a non-resisting medium of the same specific gravity, will be proportional to the arcs described in oscillating nearly. For let equal pendulums in a re sisting medium describe the unequal arcs A, B ; and the resistance of the body in the arc A will be to the resist ance of the body in the correspondent part of the arc B in the duplicate ra tio of the velocities, that is, as, A A to BB nearly. If the resistance in the arc B were to the resistance in the arc A as AB to AA, the times in the arcs A and B would be equal (by the last Prop.) Therefore the resistance AA in the arc A, or AB in the arc B, causes the excess of the time in the arc A above the time in a non-resisting medium ; and the resistance BB causes the excess of the time in the arc B above the time in a non-resisting medium. But those excesses are as the efficient forces AB and BB nearly, that is, as the arcs A and B. Q.E.D. COR, 1. Hence from the times of the oscillations in unequal arcs in a resisting medium, may be knowrn the times of the oscillations in a non- re sisting medium of the same specific gravity. For the difference of the times will be to the excess of the time in the lesser arc above the time in a non-resisting medium as the difference of the arcs to the lesser arc. COR. 2. The shorter oscillations are more isochronal, and very short ones are performed nearly in the same times as in a non-resisting medium. But the times of those which are performed in greater arcs are a little greater, because the resistance in the descent of the body, by which the time is prolonged, is greater, in proportion to the length described in the descent than the resistance in the subsequent ascent, by which the time is contracted. But the time of the oscillations, both short arid long, seems to be prolonged in some measure by the motion of the medium. For retard ed bodies are resisted somewhat less in proportion to the velocity, and ac celerated bodies somewhat more than those that proceed uniformly forwards ; SEC. VI.] OF NATURAL PHILOSOPHY. 307 because the medium, by the motion it has received from the bodies, going forwards the same way with them, is more agitated in the former case, and less in the latter ; and so conspires more or less with the bodies moved. Therefore it resists the pendulums in their descent more, and in their as cent less, than in proportion to the velocity; and these two causes concur ring prolong the time. PROPOSITION XXVIII. THEOREM XXIII. If afunependvlous body, oscillating in a cycloid, be resisted in the rati > of the moments of the time, its resistance will be to the force of grav ity as the excess of the arc described in the whole descent above the arc described in the subsequent ascent to twice the length of the pen dulum. Let BC represent the arc described in the descent, Ca the arc described in the ascent, and Aa the difference of the arcs : and things remaining as they were constructed and demonstrated in Prop. XXV, the force with which the oscillating body is urged in any place D will be to the force of resistance as the arc CD to the arc CO, which is half of that difference Aa. Therefore the force with which the oscillating body is urged at the beginning or the highest point of the cycloid, that is, the force of gravity, will be to the resistance as the arc of the cycloid, be tween that highest point and lowest point C, is to the arc CO ; that is (doubling those arcs), as the whole cycloidal arc, or twice the length of the pendulum, to the arc Aa. Q.E.D. PROPOSITION XXIX. PROBLEM VI. Supposing that a body oscillating in a cycloid is resisted in a duplicate ratio of the velocity: to find the resistance in each place. Let Ba be an arc described in one entire oscillation, C the lowest point C O K O ,S P rR Q M of the cycloid, and CZ half the whole cycloidal arc, equal to the length of the pendulum ; and let it be required to find the resistance of the body is 30S THE MATHEMATICAL PRINCIPLES [BOOK 1L any place D. Cut the indefinite right line OQ in the points O, S, P, Q,, so that (erecting the perpendiculars OK, ST, PI, QE, and with the centre O, and the aysmptotcs OK, OQ, describing the hyperbola TIGE cutting the perpendiculars ST, PI, QE in T. I, and E, and through the point I drawing KF. parallel to the asymptote OQ, meeting the asymptote OK i i K, and the perpendiculars ST and QE in L and F) the hyperbolic area PIEQ may be to the hyperbolic area PITS as the arc BC, described in the descent of the body, to the arc Ca described in the ascent ; and that the area IEF may be to the area ILT as OQ to OS. Then with the perpen dicular MN cut off the hyperbolic area PINM, and let that area be to the hyperbolic area PIEQ as the arc CZ to the arc BC described in the de scent. And if the perpendicular RG cut off the hyperbolic area PIGR, which shall be to the area PIEQ as any arc CD to the arc BC described in the whole descent, the resistance in any place D will be to the force of OR gravity as the area IEF — IGH to the area PINM. For since the forces arising from gravity with which the body is urged in the places Z, B, D, a, are as the arcs CZ. CB, CD, Ca and those arcs are as the areas PINM, PIEQ, PIGR, PITS; let those areas be the exponents both of the arcs and of the forces respectively. Let DC? be a very small space described by the body in its descent : and let it be expressed r>y the very small area RGor comprehended between the parallels RG, rg ; and produce r<? to //, so that GYlhg- and RG«r may be the contemporane ous decrements of the areas IGH, PIGR. And the increment Gllhg — IEF, or Rr X HG — -^ IEF, of the area ~ IEF —IGH will be , OQ OQ IFF to the decrement RG°r, or Rr X RG, of the area PIGR, as HG — - - OR to RG ; and therefore as OR X HG — IEF to OR X GR or OP X PL that is (because of the equal quantities OR X HG, OR X HR — OR X GR, ORHK — OPIK, PIHR and PIGR + IGH), as PIGR + IGH — OR OR IEF to OPIK. Therefore if the area - IEF — IGH be called OQ Y, and RGgr the decrement of the area PIGR be given, the increment of the area Y will be as PIGR — Y. Then if V represent the force arising from the gravity, proportional to the arc CD to be described, by which the body is acted upon in D, and R be put for the resistance, V — R will be the whole force with which the body is urged in D. Therefore the increment of the velocity is as V — R and the particle of time in which it is generated conjunctly. But the ve locity itself is as the contempo] aueous increment of the space described di- SEC. VI.J OF NATURAL PHILOSOPHY. 309 rectly and the same particle of time inversely. Therefore, since the re sistance is, by the supposition, as the square of the velocity, the increment of the resistance will (by Lem. II) be as the velocity and the increment of the velocity conjunctly, that is, as the moment of the space and V — R conjunctly ; and, therefore, if the moment of the space be given, as V — 11 ; that is, if for the force V we put its exponent PIGR, and the resist ance R be expressed by any other area Z; as PIGR — Z.v Therefore the area PIGR uniformly decreasing by the subduction of given moments, the area Y increases in proportion of PIGR — Y, and the area Z in proportion of PIGR — Z. And therefore if the areas Y and Z begin together, and at the beginning are equal, these, by the addition of equal moments, will continue to be equal • and in like man ner decreasing by equal moments, \vill vanish together. And, vice versa, if they together begin and vanish, they will have equal moments and te always equal ; and that, because if the resistance Z be augmented, the ve locity together with the arc C«, described in the ascent of the body, will be diminished ; and the point in which all the motion together with the re sistance ceases coming nearer to the point C, the resistance vanishes sooner than the area Y. And the contrary will happen when the resistance is diminished. Now the area Z begins and ends where the resistance is nothing, that is, at the beginning of the motion where the arc CD is equal to the arc CB, K /IK O S P /~R Q M and the right line RG falls upon the right line Q.E ; and at the end of the motion where the arc CD is equal to the arc Ca, and RG falls upon the right line ST. And the area* Y or — IEF — IGH begins and ends also where the resistance is nothing, and therefore where IEF and IGH are equal ; that is (by the construction), where the right line RG falls successively upon the right lines Q,E and ST. Therefore those areas begin and vanish together, and are therefore always equal. Therefore the area OR IEF — IGH is equal to the area Z, by which the resistance is ex pressed, and therefore is to the area PINM, by which the gravity is ex pressed, as the resistance to the gravity. Q.E.D. 310 THE MATHEMATICAL PRINCIPLES [BOOK 11. COR. 1 . Therefore the resistance in the lowest place C is to the force OP of gravity as the area ^ ~ IEF to the area PINM. COR. 2. But it becomes greatest where the area PIHR is to the area IEF as OR to OQ. For in that case its moment (that is, PIGR — Y) becomes nothing. COR. 3. Hence also may be known the velocity in each place, as being in the subduplicate ratio of the resistance, and at the beginning of the mo tion equal to the velocity of the body oscillating in the same cycloid with out any resistance. However, by reason of the difficulty of the calculation by which the re sistance and the velocity are found by this Proposition, we have thought fit to subjoin the Proposition following. PROPOSITION XXX. THEOREM XXIV. If a right line aB be equal to the arc of a cycloid which an oscillating body describes, and at each of its points D the perpendiculars DK be erected, which shall be to the length of the pendulum as the resistance of the body in the corresponding points of the arc to the force of grav ity ; I say, that the difference between the arc described in the whole descent and the arc described in the whole subsequent ascent drawn into half the sum of the same arcs will be equal to the area BKa which all those perpendiculars take up. Let the arc of the cycloid, de scribed in one entire oscillation, be expressed by the right line aB, equal to it, and the arc which would have been described in vaciw by the length AB. Bisect AB in C, and the point C will represent the lowest point of the cycloid, and CD Mill be as the force arising from gravity, with which the body in D i,s urged in the direction of the tangent of the cycloid, and will have the same ratio to the length of the pendulum as the force in D has to the force of gravity. Let that force, therefore, be expressed by that length CD, and the force of gravity by the length of the pendulum ; and if in DE you take DK in the same ratio to the length of the pendulum as the resistance has to the gravity, DK will be the exponent of the resistance. From the centre C with the interval CA or CB describe a semi-circle BEeA. Let the body describe, in the least time, the space Dd ; and, erecting the per pendiculars DE, de, meeting the circumference in E and e, they will be as the velocities which the body descending in vacuo from the point B would acquire in the places D and d. This appears by Prop, LII, Book L Let SEC. VLJ OF NATURAL PHILOSOPHY. 311 therefore, these velocities be expressed by those perpendiculars DE, de ; arid let DF be the velocity which it acquires in D by falling from B in the resisting medium. And if from the centre C with the interval OF we describe the circle F/M meeting the right lines de and AB in / and M, then M will be the place to which it would thenceforward, without farther resistance, ascend, and (//"the velocity it would acquire in d. Whence, also, if FO- represent the moment of the velocity which the body D, in de scribing the least space DC/, loses by the resistance of the medium ; and CN be taken equal to Cg ; then will N be the place to which the body, if it met no farther resistance, would thenceforward ascend, and MN will be the decrement of the ascent arising from the loss of that velocity. Draw F/n perpendicular to dft and the decrement Fg of the velocity DF gener ated by the resistance DK will be to the increment fm of the same velo city, generated by the force CD, as the generating force DK to the gener ating force CD. But because of the similar triangles F////, Fhg, FDC, fm is to Fm or Dd as CD to DF ; and, ex ceqno, Fg to Dd as DK to DF. Also Fh is to Fg as DF to CF ; and, ex ax/uo perturbate, Fh or MN to Do1 as DK to CF or CM ; and therefore the sum of all the MN X CM will be equal to the sum of all the Dd X DK. At the moveable point M suppose always a rectangular ordinate erected equal to the inde terminate CM, which by a continual motion is drawn into the whole length Aa ; and the trapezium described by that motion, or its equal, the rectangle Aa X |aB, will be equal to the sum of all the MN X CM, and therefore to the sum of all the Dd X DK, that is, to the area BKVTa O.E.D. COR. Hence from the law of resistance, and the difference Aa of the arcs Ca} CB, may be collected the proportion of the resistance to the grav ity nearly. For if the resistance DK be uniform, the figure BKTa will be a rec tangle under Ba and DK; and thence the rectangle under ^Ba and Aa will be equal to the rectangle under Ba and DK, and DK will be equal to jAa. Wherefore since DK is the exponent of the resistance, and the length of the pendulum the exponent of the gravity, the resistance will be to the gravity as \Aa to the length of the pendulum ; altogether as in Prop. XXVIII is demonstrated. If the resistance be as the velocity, the figure BKTa will be nearly an ellipsis. For if a body, in a non-resisting medium, by one entire oscilla tion, should describe the length BA, the velocity in any place D would be as the ordinate DE of the circle described on the diameter AB. There fore since Ea in the resisting medium, and BA in the non-resisting one, are described nearly in the same times ; and therefore the velocities in each of the points of Ba are to the velocities in the correspondent points of the length BA. nearly as Ba is to BA , the velocity in the point D in the re- 312 THE MATHEMATICAL PRINCIPLES [BJOK 11. sisting medium will be as the ordinate of the circle or ellipsis described upon the diameter Ba ; and therefore the figure BKVTa will be nearly ac ellipsis. Since the resistance is supposed proportional to the velocity, le\ OV be the exponent of the resistance in the middle point O ; and an ellip sis BRVSa described with the centre O, and the semi-axes OB, OV, will be nearly equal to the figure BKVTa, and to its equal the rectangle Act X BO. Therefore Aa X BO is to OV X BO as the area of this ellipsis to OV X BO; that is, Aa is to OV as the area of the semi-circle to the square of the radius, or as 1 1 to 7 nearly ; and, therefore, T7T Aa is to the length of the pendulum as the resistance of the oscillating body in O to its gravity. Now if the resistance DK be in the duplicate ratio of the velocity, the figure BKVTa will be almost a parabola having V for its vertex arid OV for its axis, and therefore will be nearly equal to the rectangle under f Ba and OV. Therefore the rectangle under |Ba and Aa is equal to the rec tangle f Ba X OV, and therefore OV is equal to f Aa ; and therefore the resistance in O made to the oscillating body is to its gravity as f Aa to the length of the pendulum. And I take these conclusions to be accurate enough for practical uses. For since an ellipsis or parabola BRVSa falls in with the figure BKVTa in the middle point V, that figure, if greater towards the part BRV or VSa than the other, is less towards the contrary part, and is therefore nearly equal to it. PROPOSITION XXXI. THEOREM XXV. If the resistance made to an oscillating body in each of the proportional parts of the arcs described be augmented or diminished in, a given ra tio, the difference between the arc described in the descent and the arc described in the subsequent ascent ivill be augmented or diminished in the same ratio. For that difference arises from the retardation of the pendulum by the resistance of the medium, and therefore is as the whole re tardation and the retarding resist ance proportional thereto. In the foregoing Proposition the rectan- M isr u c o «.-/ n P gle under the right line ^aB and the difference Aa of the arcs CB, Ca, was equal to the area BKTa, And that area, if the length aB remains, is augmented or diminished in the ra tio of the ordinates DK ; that is, in the ratio of the resistance and is there fore as the length aB and the resistance conjunctly. And therefore the rectangle under Aa and |aB is as aB and the resistance conjunctly, anc therefore Aa is as the resistance. QJE.D. SEC. VI.l OF NATURAL PHILOSOPHY. 313 COR. 1. Hence if the resistance be as the velocity, the difference of the arts in the same medium will be as the whole arc described : and the contrary. COR. 2. If the resistance be in the duplicate ratio of the velocity, that difference will be in the duplicate ratio of the whole arc : and the contrary. COR. 3. And universally, if the resistance be in the triplicate or any other ratio of the velocity, the difference will be in the same ratio of the. whole arc : and the contrary. COR. 4. If the resistance be partly in the simple ratio of the velocity, and partly in the duplicate ratio of the same, the difference will be partly in the ratio of the whole arc, and partly in the duplicate ratio of it: and the contrary. So that the law arid ratio of the resistance will be the same for the velocity as the law and ratio of that difference for the length of the arc. COR. 5. And therefore if a pendulum describe successively unequal arcs, and we can find the ratio of the increment or decrement of this difference for the length of the arc described, there will be had also the ratio of the increment or decrement of the resistance for a greater or less velocity. GENERAL SCHOLIUM. From these propositions we may find the resistance of mediums by pen dulums oscillating therein. I found the resistance of the air by the fol lowing experiments. I suspended a wooden globe or ball weighing oT^ ounces troy, its diameter CJ London inches, by a fine thread on a firm hook, so that the distance between the hook and the centre of oscillation of the globe was 10| feet. I marked on the thread a point 10 feet and 1 inch distant from the centre of suspension • and even with that point I placed a ruler divided into inches, by the help whereof I observed the lengths of the arcs described by the pendulum. Then I numbered the oscillations ia which the globe would lose -{- part of its motion. If the pendulum was drawn aside from the perpendicular to the distance of 2 inches, and thence let go, so that in its whole descent it described an arc of 2 inches, and in the first whole oscillation, compounded of the descent and subsequent ascent, an arc of almost 4 inches, the same in 164 oscillations lost j part of its motion, so as in its last ascent to describe an arc of If inches. If in the first descent it described an arc of 4 inches, it lost j part of its mo tion in 121 oscillations, so as in its last ascent to describe an arc of 3^ inches. If in the first descent it described an arc of 8, 16, 32, or 64 inches, it lost | part of its motion in 69, 35|, 18|-7 9| oscillations, respectively. Therefore the difference between the arcs described in the first descent and the last ascent was in the 1st, 2d, 3d, 4th, 5th, 6th cases, }, 1. 1, 2, 4, 8 inches respectively. Divide those differences by the number of oscillations in each case, and in one mean oscillation, wherein an arc of 3£ , 7-|, 15, 30 314 THE MATHEMATICAL PRINCIPLES [BOOK Jl. 60, 120 inches was described, the difference of the arcs described in the descent and subsequent ascent will be ¥|^, ^{^ e\> T4r; -sji fir parts of an inch, respectively. But these differences in the greater oscillations are in the duplicate ratio of the arcs described nearly, but in lesser oscillations something greater than in that ratio ; and therefore (by Cor. 2, Prop. XXXI of this Book) the resistance of the globe, when it moves very swift, is in the duplicate ratio of the velocity, nearly; and when it moves slowly, somewhat greater than in that ratio. Now let V represent the greatest velocity in any oscillation, and let A, B, and C be given quantities, and let us suppose the difference of the arcs 3^ to be AV + BV2 + CV2. Since the greatest velocities are in the cycloid as ^ the arcs described in oscillating, and in the circle as | the chords of those arcs ; and therefore in equal arcs are greater in the cycloid than in the circle in the ratio of | the arcs to their chords ; but the times in the circle are greater than in the cycloid, in a reciprocal ratio of the velocity ; it is plain that the differences of the arcs (which are as the resistance and the square of the time conjunctly) are nearly the same in both curves : for in the cycloid those differences must be on the one hand augmented, with the resistance, in about the duplicate ratio of the arc to the chord, because of the velocity augmented in the simple ratio of the same ; and on the other hand diminished, with the square of the time, in the same duplicate ratio. Therefore to reduce these observations to the cycloid, we must take the same differences of the arcs as were observed in the circle, and suppose the greatest velocities analogous to the half, or the whole arcs, that is, to the numbers ±, 1, 2, 4, 8, 16. Therefore in the 2d, 4th, and 6th cases, put 1,4, and 1 6 for V ; and the difference of the arcs in the 2d case will become i 2 * = A + B + C; in the4th case, ^- = 4A + SB + 160 ; in the 6th 121 OOj case, ^ = 16A + 64B -f- 256C. These equations reduced give A = 9? 0,000091 6, B =-. 0,0010847, and C = 0,0029558. Therefore the difference of the arcs is as 0,0000916V -f 0,0010847V* + 0,0029558 V* : and there fore since (by Cor. Prop. XXX, applied to this case) the re.-ist;mcc of the globe in the middle of the arc described in oscillating, where the velocity is V, is to its weight as T7TAV -f- T\BV^ + f CV2 to the length of the pendulum, if for A, B, and C you put the numbers found, the resistance of the globe will be to its weight as 0,0000583V + 0,0007593V* + 0,OJ22169V2 to the length of the pendulum between the centre of suspension and the ruler, that is, to 121 inches. Therefore since V in the second case repre sents 1, in the 4th case 4, and in the 6th case 16, the resistance will be to the weight of the globe, in the 2d case, as 0,0030345 to 121 ; in the 4th, as 0,041748 to 121 ; in the 6th, as 0,61705 to 121. SEC. VI.] OF NATURAL PHILOSOPHY. 315 The arc, which the point marked in the thread described in the 6th case, was of 120 — Q^, or 119/g inches. And therefore since the radius was ya 121 inches, and the length of the pendulum between the point of suspen sion and the centre of the globe was 126 inches, the arc which the centre of the globe described was 124/T inches. Because the greatest velocity of the oscillating body, by reason of the resistance of the air, does not fall on the lowest point of the arc described, but near the middle place of the whole arc, this velocity will be nearly the same as if the globe in its whole descent in a non-resisting medium should describe 62 ^ inches, the half of that arc, and that in a cycloid, to which we have above reduced the motion of the pendulum; and therefore that velocity will be equal to that which the globe would acquire by falling perpendicularly from a height equal to the versed sine of that arc. But that versed sine in the cycloid is to that arc 62/2 as the same arc to twice the length of the pendulum 252, and there fore equal to 15,278 inches. Therefore the velocity of the pendulum is the same which a body would acquire by falling, and in its fall describing a space of 15,278 inches. Therefore with such a velocity the globe meets with a resistance which is to its weight as 0,61705 to 121, or (if we take that part only of the resistance which is in the duplicate ratio of the ve- loc.ty) as 0,56752 to 121. I found, by an hydrostatical experiment, that the weight of this wooden globe was to the weight of a globe of water of the same magnitude as 55 to 97: and therefore since 121 is to 213,4 in the same ratio, the resistance made to this globe of water, moving forwards with the above-mentioned velocity, will be to its weight as 0,56752 to 213,4, that is, as 1 to 376^. Whence since the weight of a globe of water, in the time in which the globe with a velocity uniformly continued describes a length of 30,556 inches, will generate all that velocity in the falling globe, it is manifest that the force of resistance uniformly continued in the same time will take away a velocity, which will be less than the other in the ratio of 1 to 376^-0, that is, the rr^-r part of the whole velocity. And therefore in the time 37 VSG Jiat the globe, with the same velocity uniformly continued, would describe the length of its semi-diameter, or 3T\ inches, it would lose the 3^42 part of its motion. I also counted the oscillations in which the pendulum lost j part of its motion. In the following table the upper numbers denote the length of the arc described in the first descent, expressed in inches and parts of an inch ; the middle numbers denote the length of the arc described in the last as cent ; and in the lowest place are the numbers of the oscillations. I give un account of this experiment, as being more accurate than that in which 316 THE MATHEMATICAL PRINCIPLES [BOOK ll only 1 part of the motion was lost. I leave the calculation to such as are disposed to make it. First descent ... 2 4 8 16 32 64 Last ascent . . , 1| 3 6 12 24 48 Numb.ofoscilL . .374 272 162i 83J 41f 22| I afterward suspended a leaden globe of 2 inches in diameter, weighing 26 1 ounces troy by the same thread, so that between the centre of the globe and the point of suspension there was an interval of 10^ feet, and 1 counted the oscillations in which a given part of the motion was lost. The iirst of the following tables exhibits the number of oscillations in which J- part of the whole motion was lost ; the second the number of oscillations in which there was lost \ part of the same. First descent .... 1 2 4 8 16 32 64 Last ascent .... f ' J 3^ 7 14 28 56 Numb, of oscilL . . 226 228 193 140 90^ 53 30 First descent .... 1 2 4 8 16 32 64 Last ascent .... £ 1^ 3 6 12 24 4S Nunib. of oscill. . .510 518^ 420 318 204 12170 Selecting in the first table the 3d, 5th, and 7th observations, and express ing the greatest velocities in these observations particularly by the num bers 1, 4, 16 respectively, 'and generally by the quantity V as above, there will come out in ihe 3d observation ~- = A + B + C, in the 5th obser- 2 8 vation ^— = 4A 4- 8B + 16C. in the 7th observation ^-- == 16A 4- 64B t- ,t(j j oU 256C. These equations reduced give A = 0,001414, B == 0,000297, C — 0,000879. And thence the resistance of the globe moving with the velocity V will be to its weight 26^ ounces in the same ratio as 0,0009V + 0,000208V* + 0,000659V2 to 121 inches, the length of the pendulum. And if we regard that part only of the resistance which is in the dupli cate ratio of the velocity, it will be to the weight of the globe as 0,000659V2 to 121 inches. But this part of the resistance in the first experiment was to the weight oi' the wooden globe of 572-72 ounces as 0,002217V2 to 121 ; and thence the resistance of the wooden globe is to the resistance of the leaden one (their velocities being equal) as 57/2- into 0,002217 to 26 J- into 0,000659, that is, as 7|- to 1. The diameters of the two globes were 6f and 2 inches, and the squares of these are to each other as 47 £ and 4, or 11-J-f and 1, nearly. Therefore the resistances of these equally swift globes were in less than a duplicate ratio of the diameters. But we have not yet considered the resistance of the thread, which was certainly very considerable, and ought to be subducted from the resistance of the pendu lums here found. I could not determine this accurately, but I found il SEC. VI.J OF NATURAL PHILOSOPHY. 3 1/ greater than a third part of the whole resistance of the lesser pendulum ; and thence I gathered that the resistances of the globes, when the resist ance of the thread is subducted, are nearly in the duplicate ratio of their diameters. For the ratio of 7} — } to 1 — £, or l(H to 1 is not very different from the duplicate ratio of the diameters 1 L}f to I. Since the resistance of the thread is of less moment in greater globes, I tried the experiment also with a globe whose diameter was ISf inches. The length of the pendulum between the point of suspension and the cen tre of oscillation was 122| inches, and between the point of suspension and the knot in the thread 109| inches. The arc described by the knot at the first descent of the pendulum was 32 inches. The arc described by the same knot in the last ascent after five oscillations was 2S inches. The sum of the arcs, or the whole arc described in one mean oscillation, was 60 inches. The difference of the arcs 4 inches. The y1,,- part of this, or the difference between the descent and ascent in one mean oscillation, is f of an inch. Then as the radius 10(J| to the radius 122^, so is the whole arc of 60 inches described by the knot in one mean oscillation to the whole arc of 67} inches described by the centre of the globe in one mean oscillation ; and so is the difference | to a new difference 0,4475. If the length of the arc described were to remain, and the length of the pendulum should be augmented in the ratio of 126 to 122}, the time of the oscillation would be augmented, and the velocity of the pendulum would be diminished in the subduplicate of that ratio ; so that the difference 0,4475 of the arcs de scribed in the descent and subsequent ascent would remain. Then if the arc described be augmented in the ratio of 124 33T to 67}, that difference 0.4475 would be augmented in the duplicate of that ratio, and so would become 1,5295. These things would be so upon the supposition that the resistance of the pendulum were in the duplicate ratio of the velocity. Therefore if the pendulum describe the whole arc of 12433T inches, and its length between the point of suspension and the centre of oscillation be 126 inches, the difference of the arcs described in the descent and subsequent ascent would be 1,5295 inches. And this difference multiplied into the weight of the pendulous globe, which was 208 ounces, produces 318,136. Again ; in the pendulum above-mentioned, made of a wooden globe, when its centre of oscillation, being 126 inches from the point of suspension, de scribed the whole arc of 124 /T inches, the difference of the arcs described in the descent and ascent was ^^ into ^. This multiplied into the i/wi y^ weight of the globe, which was 57-272 ounces, produces 49,396. But I mul tiply these differences into the weights of the globes, in order to find their resistances. For the differences arise from the resistances, and are as the resistances directly and the weights inversely. Therefore the resistances are as the numbers 318,136 and 49,396. But that part of the resistance 31 S THE MATHEMATICAL PRINCIPLES [BOOK 1L of the lesser globe, which is in the duplicate ratio of the velocity, was to the whole resistance as 0,56752 to- 0,61675, that is, as 45,453 to 49,396 ; whereas that part of the resistance of the greater globe is almost equal to its whole resistance ; and so those parts are nearly as 318,136 and 45,453, that is, as 7 and 1. But the diameters of the globes are 18f and 6| ; and their squares 351T9¥ and 47 £J are as 7,438 and 1, that is, as the resistances of the globes 7 and 1, nearly. The difference of these ratios is scarce greater than may arise from the resistance of the thread. Therefore those parts of the resistances which are, when the globes are equal, as the squares of the velocities, are also, when the velocities are equal, as the squares of the diameters of the globes. But the greatest of the globes I used in these experiments was not per fectly spherical, and therefore in this calculation I have, for brevity's sake, neglected some little niceties ; being not very solicitous for an accurate calculus in an experiment that was not very accurate. So that I could wish that these experiments were tried again with other globes, of a larger size, more in number, and more accurately formed ; since the demonstra tion of a vacuum depends thereon. If the globes be taken in a geometrical proportion, as suppose whose diameters are 4, 8, 16, 32 inches; one may collect from the progression observed in the experiments what would hap pen if the globes were still larger. In order to compare the resistances of different fluids with each other, 1 made the following trials. I procured a wooden vessel 4 feet long, 1 foot broad, and 1 foot high. This vessel, being uncovered, 1 filled with spring water, and, having immersed pendulums therein, I made them oscillate in the water. And I found that a leaden globe weighing 166| ounces, and in diameter 3f inches, moved therein as it is set down in the following table ; the length of the pendulum from the point of suspension to a certain point marked in the thread being 126 inches, and to the centre of oscilla tion 134 f inches. The arc described in } the first descent, by a point marked in \ 64 . 32 . 16 . $ . 4 . 2 . 1 . £ . J the thread was \ inches. The arc described in ) the last ascent was V 48 . 24 . 12 . 6 . 3 . 1| . £ . f . T\ inches. \ The difference of the arcs, proportional to the 'motion lost, was inches. The number of the os cillations in water. The number of the os cillations in air. 16 «. li . 3 . 7 . lH.12f.13j 85i . 287 . 535 SEC. VI.] OF NATURAL PHILOSOPHY. 319 In the experiments of the 4th column there were equal motions lost in 535 oscillations made in the air, and If in water. The oscillations in the air were indeed a little swifter than those in the water. But if the oscil lations in the water were accelerated in such a ratio that the motions of the pendulums might be equally swift in both mediums, there would be still the same number 1 j of oscillations in the water, and by these the same quantity of motion would be lost as before ; because the resistance i> increased, and the square of the time diminished in the same duplicate ra tio. The pendulums, therefore, being of equal velocities, there were equal motions lost in 535 oscillations in the air, and 1} in the water; and there fore the resistance of the pendulum in the water is to its resistance in the air as 535 to 1 }. This is the proportion of the whole resistances in the case of the 4th column. Now let AV + CV2 represent the difference of the arcs described in the descent and subsequent ascent by the globe moving in air with the greatest velocity V ; and since the greatest velocity is in the case of the 4th column to the greatest velocity in the case of the 1st column as 1 to 8 ; and that difference of the arcs in the case of the 4th column to the difference in the 2 16 case of the 1st column as ^ to 7, or as 86 J to 4280 ; put in these cases 1 and 8 for the velocities, and 85 1 and 4280 for the differences of the arcs, and A + C will be — S5|, and 8A -f 640 ==• 4280 or A + SC = 535 ; and then by reducing these equations, there will come out TC = 449^ and C = 64T\ and A = 21f ; and therefore the resistance, which is as TVAV + fCV2, will become as 13T6TV + 48/^Y2. Therefore in the case of the 4th column, where the velocity was 1, the whole resistance is to its part proportional to the square of the velocity as 13T6T + 48/F or 61 }f to 48/e ; and therefore the resistance of the pendulum in water is to that part of the resistance in air, which is proportional to the square of the velocity, and which in swift motions is the only part that deserves consid eration, as 61}^ to 4S/g and 535 to 1} conjunctly, that is, as 571 to 1. If the whole thread of the pendulum oscillating in the water had been im mersed, its resistance would have been still greater ; so that the resistance of the pendulum oscillating in the water, that is, that part which is pro portional to the square of the velocity, and which only needs to be consid ered in swift bodies, is to the resistance of the same whole pendulum, oscil lating in air with the same velocity, as about 850 to 1, that is as, the den sity of water to the density of air, nearly. In this calculation we ought also to have taken in that part of the re sistance of the pendulum in the water which was as the square of the ve locity ; but I found (which will perhaps seem strange) that the resistance in the water was augmented in more than a duplicate ratio of the velocity. In searching after the cause, I thought upon this, that the vessel was toe 320 THE MATHEMATICAL PRINCIPLES [BOOK II. narrow for the magnitude of the pendulous globe, and by its narrowness obstructed the motion of the water as it yielded to the oscillating globe. For when I immersed a pendulous globe, whose diameter was one inch only, the resistance was augmented nearly in a duplicate ratio of the velocity, I tried this by making a pendulum of two globes, of which the lesser and lower oscillated in the water, and the greater and higher was fastened to the thread just above the water, and, by oscillating in the air, assisted the motion of the pendulum, and continued it longer. The experiments made by this contrivance proved according to the following table. Arc descr. in first descent . .16.8. 4. 2.1.1.1 Arc descr. in last ascent . . 12 . 6 . 3 . li . J . | . T3F Dif. of arcs, proport. to 1 . pi i motion lost $ T • «r • T* Number of oscillations... 3f . 6j . 12^. 211 . 34 . 53 . 62) In comparing the resistances of the mediums with each other, I also caused iron pendulums to oscillate in quicksilver. The length of the iron wire was about 3 feet, and the diameter of the pendulous globe about i of an inch. To the wire, just above the quicksilver, there was fixed another leaden globe of a bigness sufficient to continue the motion of the pendulum for some time. Then a vessel, that would hold about 3 pounds of quick silver, was filled by turns with quicksilver and common water, that, by making the pendulum oscillate successively in these two different fluids, I might find the proportion of their resistances ; and the resistance of the quicksilver proved to be to the resistance of water as about 13 or 14 to 1 ; that is. as the density of quicksilver to the density of water. When I made use of a pendulous globe something bigger, as of one whose diameter was about ^ or | of an inch, the resistance of the quicksilver proved to be to the resistance of the water as about 12 or 10 to 1. But the former experi ment is more to be relied on, because in the latter the vessel was too nar row in proportion to the magnitude of the immersed globe; for the vessel ought to have been enlarged together with the globe. I intended to have repeated these experiments with larger vessels, and in melted metals, and other liquors both cold and hot ; but I had not leisure to try all: and be sides, from what is already described, it appears sufficiently that the resist ance of bodies moving swiftly is nearly proportional to the densities of the fluids in which they move. I do not say accurately ; for more tena cious fluids, of equal density, will undoubtedly resist more than those that are more liquid ; as cold oil more than warm, warm oil more than rain water, and water more than spirit of wine. But in liquors, which are sen sibly fluid enough, as in air, in salt and fresh water, in spirit of wine, of turpentine, and salts, in oil cleared of its fseces by distillation and warmed, in oil of vitriol, and in mercury, and melted metals, and any other such like, that are fluid enough to retaia for some time the motion impressed SEC. VI.J OF NATURAL PHILOSOPHY. 321 upon them by the agitation of the vessel, and which being poured out are easily resolved into drops, I doubt not but the rule already laid down may be accurate enough, especially if the experiments be made with larger pendulous bodies and more swiftly moved. Lastly, since it is the opinion of some that there is a certain ^ethereal medium extremely rare and subtile, which freely pervades the pores of all bodies ; and from such a medium, so pervading the pores of bodies, some re sistance must needs arise; in order to try whether the resistance, which wre experience in bodies in motion, be made upon their outward superficies only, or whether their internal parts meet with any considerable resistance upon their superficies, I thought of the following experiment I suspended a round deal box by a thread 11 feet long, on a steel hook, by means of a ring of the s-ime metal, so as to make a pendulum of the aforesaid length. The hook had a sharp hollowr edge on its upper part, so that the upper arc of the ring pressing on the edge might move the more freely ; and the thread was fastened to the lower arc of the ring. The pendulum being thus pre pared, I drew it aside from the perpendicular to the distance of about 6 feet, and that in a plane perpendicular to the edge of the hook, lest the ring, while the pendulum oscillated, should slide to and fro on the edge of the hook : for the point of suspension, in which the ring touches the hook, ought to remain immovable. I therefore accurately noted the place to which the pendulum was brought, and letting it go, I marked three other places, to which it returned at the end of the 1st, 2d, and 3d oscillation. This I often repeated, that I might find those places as accurately as pos sible. Then I filled the box with lead and other heavy metals that were near at hand. But, first, I weighed the box when empty, and that pnrt of the thread that went round it, and half the remaining part, extended be tween the hook and the suspended box ; for the thread so extended always acts upon the pendulum, when drawn aside from the perpendicular, with half its weight. To this weight I added the weight of the air contained in the box And this whole weight was about -fj of the weight of the box when filled writh the metals. Then because the box when full of the metals, by ex tending the thread with its weight, increased the length of the pendulum, f shortened the thread so as to make the length of the pendulum, when os cillating, the same as before. Then drawing aside the pendulum to the place first marked, and letting it go, I reckoned about 77 oscillations before the box returned to the second mark, and as many afterwards before it came to the third mark, and as many after that before it came to the fourth xnark. From whence I conclude that the whole resistance of the box, when full, had not a greater proportion to the resistance of the box, when empty, than 78 to 77. For if their resistances were equal, the box, when full, by reason of its vis insita, which was 78 times greater than the vis tfuritoof the same when empty, ought to have continued its oscillating motion so 21 322 THE MATHEMATICAL PRINCIPLES | BOOK II. much the longer, and therefore to have returned to those marks at the end of 78 oscillations. But it returned to them at the end of 77 oscillations. Let, therefore, A represent the resistance of the box upon its external superficies, and B the resistance of the empty box on its internal superficies ; and if the resistances to the internal parts of bodies equally swift be as the matter, or the number of particles that are resisted, then 78B will be the resistance made to the internal parts of the box, when full ; and therefore the whole resistance A + B of the empty box will be to the whole resist ance A + 7SB of the full box as 77 to 78, and, by division, A + B to 77B as 77 to 1 ; and thence A + B to B as 77 X 77 to 1, and, by division again, A to B as 5928 to 1. Therefore the resistance of the empty box in its internal parts will be above 5000 times less than the resistance on its external superficies. This reasoning depends upon the supposition that the greater resistance of the full box arises not from any other latent cause, but only from the action of some subtile fluid upon the included metal. This experiment is related by memory, the paper being lost in which I had described it ; so that I have been obliged to omit some fractional parts, which are slipt out of my memory ; and I have no leisure to try it again. The first time I made it, the hook being weak, the full box was retarded sooner. The cause I found to be, that the hook was not strong enough to bear the weight of the box : so that, as it oscillated to and fro, the hook was bent sometimes this and sometimes that way. I therefore procured a hook of sufficient strength, so that the point of suspension might remain unmoved, and then all things happened as is above described. SEC. VI I.] OF NATURAL PHILOSOPHY. 323 SECTION VII. Of the, motion of fluids, and the resistance made to projected bodies. PROPOSITION XXXII. THEOREM XXVI. Suppose two similar systems of bodies consisting of an equal number of particles, and let the correspondent particles be similar and propor tional, each in, one system to each in the other, and have a like situa tion among themselves, and the same given ratio of density to each other ; and let them begin to move anwng the